I 


Louis  Wedle  Go. 

MAUTICAL  msTRHMBSTO, 

CHARTS  &  B(H)K& 

6  CaUfornla  St 

sail  FaiNcisco,  calipopjiia 


UNITED  STATES  IIYDKOGKAPHIC  OFFICE. 

No.  yo: 


THE  DEVELOPMENT 


OF 


GREAT  CIRCLE  SAILING-, 


BY 


Gr.    ^V.    X.-ITTLEHALES, 

U.    S.     HYDROGRAPHIC    OFFICE. 


SECONJ>    KDITION. 


WASHINGTON: 

GOVERNMENT    PRINTING    OFFICE. 
1899. 


c  ()  isr  T  1^]  N  T  s . 


Page. 

Prekack  to  the  Fikst  P^dition 5 

Prei'ack  To  tiik  Skcond  Edition 7 

Introduction v 9 

Section  I. — Methods  l!eqnirin<j  a  Knowledge  of  tJw  I'ertex. 

Towson's  Method 12 

Deichm.iii's  Metliod 13 

Brevoort's  Method 11 

Bcrj^en's  Method 15 

The  Terrestrial  Globe 19 

The  Direct  Track  Scale 19 

The  Measurement  of  Courses  and  Distances  on  the  Direct  Track  Scale 20 

Section  II. — Methods  Dependintj  upon  the  (Inomonic  Projection. 

Godfray's  Gnomonic  Chart 22 

Godfray's  Course  and  Distance  Diagram 21 

Knorr's  (inomouic  Chart 25 

Iliileret's  Gnomonic  Chart 25 

Jenzcn's  Gnomonic  Chart 26 

Herrle's  Method 27 

Measurement  of  Great  Circle  Distances  on  Gnomonic  Charts 28 

Measurement  of  Great  Circle  Courses  on  Gnomonic  C'harts 30 

,  Section  III. — Miscellaneous  Methods. 

Airy's  Method 31 

Fisher's  Method  for  Circular  Arc  Sailing 35 

Chauvenet's  Great  Circle  Protractor 36 

Sigsbee's  Great  Circle  Protractor 38 

Harris's  Method 41 

Proctor's  Method 45 

Spherical  Traverse  Tables 50 

Great  Circle  Courses  from  the  Solar  Azimuth  Tables 50 

Section  IV. — Methods  liequiring  Compulation. 

The  C()mi)utation  of  Great  Circle  Distances 53 

The  ('omi>utation  of  the  Great  Circle  Course  and  Distance,  and  the  Latitudes 

and  Longitudes  of  Points  on  Great  C'ircular  Arcs 53 

Asnius's  Method  for  the  Construction  of  a  Great  Circle  on  the  ^lercator  Pro- 
jection    55 

Zescevich's  Method  for  Finding  the  Positions  of  the  Points  of  the  Arc  of  a 

Great  Circle 59 

The  Computation  of  the  Latitude  at  the  Middle  Longitude 62 

List  of  Literature  upon  the  Subject  of  Great  Circle  Sailing 63 

3 


PREFACE  TO  THE  FIRST  EDITION. 


Tliis  i)ablicatiou  lias  for  its  object  tlie  furtlierauce  of  tlie effort  of  tlie 
Bureau  of  Xavigatiou  of  the  Xavy  IJepartment  to  keep  i)ace  with  the 
progress  of  the  nautical  sciences.  It  consists  of  an  exposition  of 
giaphical  and  analytical  methods  embodying  cardinal  priuciples  relat- 
ing to  the  great  circle,  as  applied  to  navigation,  and  gives  publicity  for 
the  first  time  to  several  of  the  most  convenient  and  useful  methods  yet 
devised. 

The  actual  state  of  the  science  of  great  circle  sailing  is  here  pre- 
sented, so  as  to  give  a  clear  conception  of  each  method,  and  to  furnish 
references  where  more  extended  information  can  be  found. 

George  L.  Dyer, 

Hydrographer. 


PREFACE  TO  THE  SECOND  EDITION. 


During  tha  decade  that  has  elapsed  since  the  first  edition  of  tins 
publication  was  printed,  a  fuller  recognition  of  the  place  of  the  great 
circle  route  in  the  problem  of  accelerating  ocean  transit  has  stimulated 
an  advance  to  methods  by  which  great  circle  courses  cau  be  taken  from 
the  Solar  Azimuth  Tables  or  measured  from  the  chart  compass  with 
very  great  facility. 

All  the  parts  of  the  original  work  have  been  retained  in  this  edition, 
and  explanations  of  the  new  developments  have  been  incorporated  into 
those  sections  of  the  book  in  which  they  are  appropriate. 

J.  E.  Cratg, 
Captain,  U.  S.  N.,  Hydrographer. 

IlYimOGKAlMlIC    OKriCK, 

Bureau  of  Equii'ment, 

May  1,  1899. 


TIIK  DEVKLOPMEFr  OF  (IREAT  CIRCLK  SAILING. 


INTRODUCTION. 

It  is  not  seiicrally  recognized  that  science,  employing  the  mathe- 
nuitician  and  the  engineer  ali lie  in  the  problem  of  shortening  the  dura- 
tion of  ocean  transit,  has  acconiplislied  as  much  by  causing  ships  to 
travel  fewer  miles  as  by  causing  them  to  travel  faster.  In  the  age  of 
steam  propulsion  the  route  of  minimum  distance  in  ocean  trausit  is 
coming-  to  take  the  place  which  was  held  by  the  route  through  the 
regions  of  favorable  winds  in  the  age  of  sail  i)ropalsion.  A  knowledge 
of  the  i)riuciples  of  tlie  great  circle  must  have  been  coeval  with  the 
knowledge  of  the  spherical  form  of  the  earth,  but  in  the  early  days  of 
ocean  navigation  great  circle  sailing  was  doubly  impracticable,  for 
seamen  were  without  the  means  of  tinding  longitude  and-,  moreover, 
the  course  of  ships  was  controlled  by  the  wind.  Inasmuch  as  great 
circle  courses  alter  continuously  in  proceeding  along  tlie  track,  it 
becomes  necessary  to  know  the  latitude  and  longitude  of  tlie  ship  in 
order  to  determine  the  course  to  be  followed.  At  the  present  day  there 
are  convenient  means  for  determining  at  sea  the  longitude  as  well  as 
the  latitude  with  considerable  i)recisiou,  but  in  the  early  ])art  of  the 
present  century  these  means  did  not  exist  and  the  principles  of  great 
circle  sailing  could  not  be  applied. 

Besides  increasing  the  rate  of  travel,  modern  motive  power,  by 
making  possible  a  departure  from  the  old  meteorological  routes,  has 
had  another  and  a  greater  eftect  in  the  progress  of  the  universal  policy 
of  civilized  nations  to  accelerate  transit  from  place  to  place  to  the 
utmost  possible  extent,  because,  under  steam, even  if  they  go  no  taster, 
ships  may  j^et  get  I'arther  toward  the  poit  of  destination  in  a  giv^en 
time  since  they  may  be  navigated  along  arcs  of  great  circles  of  the 
earth.  The  increasing  rec'ognition  among  mariners  of  the  sound  prin- 
ciple of  conducting  a  ship  along  the  arc  of  the  great  circle  Joining  the 
points  of  departure  and  destination,  and  the  expanding  sense  of  the 
advantages  to  be  gained  by  a  knowledge  of  this  branch  of  nautical 
science,  have  greatly  heightened  the  value  of  methods  which  place  the 
benefits  of  the  knowledge  and  use  of  the  great  circle  trade  at  the  serv- 
ice of  the  mariner  without  the  labor  of  the  cahuilations  which  are  nec- 
essary to  find  the  series  of  courses  to  be  steered.     The  general  lack  of 

9 


10 

the  application  of  tlie  principles  of  the  great  circle  in  later  times,  and 
even  in  the  jireseut  generation,  seems  to  have  resulted  not  from  the 
want  of  recognizing  that  the  shortest  distance  between  any  two  places 
on  the  earth's  surface  is  the  distance  along  the  arc  of  the  great  circle 
passing  through  them,  nor  that  the  great  circle  course  is  the  only  true 
course  and  that  the  courses  in  Mercator  and  parallel  sailing  are  cir- 
cuitous, but  to  the  tedious  operations  which  have  been  necessary  and 
to  the  want  of  concise  methods  for  rendering  these  benefits  readily 
available. 

A  knowledge  of  the  great  circle  route  is  important  in  working  to 
windward  with  sailing  ships,  especially  when  far  from  the  port  of  des- 
tination. A  seaman  not  bearing  in  mind  his  great  circle  course,  which 
is  the  only  one  that  heads  the  vessel  for  her  port,  may  unwittingly  sail 
away  from  that  port  by  taking  the  wrong  tack.  The  great  circle  course 
to  a  far  distant  port  may  vary  three  or  four  points  from  the  rhumb 
course.  For  example,  on  the  route  from  Yokohama  (Cape  King)  to  Cape 
Flattery  the  great  circle  course  at  Cape  King  is  NF.,  while  the  rhumb 
course  is  E.  by  N.,  a  diiierence  of  three  points.  In  this  case  for  a  wind 
directly  ahead  on  the  rhumb  route  an  uninformed  mariner  would  lay  his 
vessel  on  either  tack  indifferently.  If  on  the  port  tack,  his  vessel  would 
head  SE.  by  S.,  nine  points  away  from  the  bearing  of  Cape  Flattery. 
On  the  starboard  tack  she  would  head  IST.  by  E.,  only  three  points  away. 
This  is  perhaps  an  extreme  case,  but  it  may  serve  to  show  how  important 
it  is  that  the  master  of  a  vessel  should  know  his  great  circle  course  even 
when  not  distinctly  i^ursuing  a  continuous  great  circle  route. 

It  has  been  i^ointed  out  that  the  rhumb  line,  although  appearing  as 
a  straight  line  on  the  Mercator  chart,  and  thus  giving  a  false  idea  that 
it  represents  the  shortest  route,  is  in  reality  a  roundabout  track,  and 
that  it  is  only  when  a  vessel's  course  is  shaped  by  the  great  circle  pass- 
ing through  the  places  of  departure  and  destination  that  she  has  the 
shortest  possible  distance  to  make  and  heads  for  her  port  as  if  it  were 
in  sight 'throughout  the  voyage. 

In  the  case  of  the  great  circle  track  between  Yokohama  and  Cape 
Flattery,  the  vertex,  or  point  of  highest  latitude  reached,  is  shown  to 
lie  within  Bering  Sea,  and  the  track  is  consequently  obstructed  by  the 
Aleutian  Islands.  It  frequently  occurs,  when  laying  out  an  extended 
great  circle  route,  that  the  lay  of  the  land,  or  the  extreme  of  climate,  or 
dangers  to  navigation,  when  weighed  in  connection  with  the  saving  of 
distance  that  is  made  good  on  the  great  circle,  leads  the  mariner  to 
limit  his  track  by  a  given  parallel  of  latitude  higher  than  which  he 
decides  not  to  go.  Under  these  conditions  the  shortest  route  to  follow 
is  made  up  of  an  arc  of  the  limiting  parallel  of  latitude  and  two  arcs 
of  great  circles  which  pass,  respectively,  through  the  points  of  depar- 
ture and  destination  and  whose  vertices,  or  points  of  highest  latitude 
reached,  lie  on  the  limiting  parallel.  In  such  a  case  the  vessel's  course 
is  laid  from  the  i)oint  of  departure  along  the  first  great  circle  arc  until 
its  highest  latitude,  which  is  the  latitude  of  the  limiting  parallel,  is 


11 

reached;  thence  along  the  limiting  parallel  to  the  point  of  the  highest 
latitude  of  the  second  great  circle;  and  finally  along  the  second  great 
circle  to  the  point  of  destination.    tSuch  a  route  is  called  composite. 

With  the  exception  of  the  miscellaneous  methods  of  Airy,  Chau- 
venet,  Harris,  Fisher,  Sigsbee,  and  Proctor,  the  development  of 
grai)liical  methods  in  this  branch  of  nautical  science  has  proceeded  in 
two  distinct  lines-,  the  history  of  one  is  the  history  of  the  development 
of  the  principles  of  the  gnomonic  projection,  and  that  of  the  other  is 
an  account  of  the  various  devices  wliich  have  been  contrived  to  find 
the  vertex  of  a  particular  great  circle. 


SECTION  I 


METHODS   EEQUIRIFG  A  KNOWLEDGE  OF   THE  VERTEX. 

By  a  system  of  great  circles  is  meant  all  great  circles  whose  common 
diameter  is  a  diameter  of  the  equator.  If  we  consider  such  a  system  of 
circles  to  be  drawn  upon  the  surface  of  a  sphere,  and  imagine  tbe  com- 
mon diameter  to  revolve  in  the  plane  of  the  equator,  all  possible  great 
circles  Avill  be  described,  for  we  have  at  first  drawn  all  the  great  circles 
of  one  system  and  then  revolved  this  system  into  all  possible  positions. 
From  these  considerations  it  is  obvious  that  a  distinguishing  feature  of 
every  great  circle  is  its  inclination  to  the  plane  of  the  equator,  or  the 
latitude  of  its  vertex.  All  properties  of  spherical  courses,  latitudes, 
and  longitudes  and  distances  from  the  vertex  of  a  certain  great  circle  of 
one  system  are  identical  in  the  corresponding  great  circle  of  every  other 
system.  Therefore,  by  tabulating  these  properties  for  each  great  circle 
of  one  system,  tables  of  the  properties  of  all  great  circles  are  formed. 

towson's  method. 

In  1850  Mr.  John  T.  Towson,  examiner  to  the  Mercantile  Marine  Board 
at  Liverpool,  proposed  a  method  for  facilitating  great  circle  sailing,  con- 
sisting in  the  use  of  tables,  such  as  described  above,  and  a  diagram,  by 
means  of  which  the  whole  operation  of  finding  the  successive  spherical 
courses  and  distances  on  any  great  circle  is  reduced  almost  to  an  affair 
of  inspection.  These  tables,  as  published  by  the  British  Admiralty, 
give  the  latitudes,  spherical  courses,  and  distances  on  great  circles  of 
the  earth  corresponding  to  each  degree  of  longitude  reckoned  from 
the  meridian  of  the  great  circle's  vertex.  Mr.  Towson's  diagram,  the 
whole  object  of  which  is  to  find  the  vertex  of  the  particular  great  circle 
which  passes  through  any  two  places,  is  constructed  as  follows: 
-p  -P 


a         I, 

Let  A  and  B  represent  two  points  on  the  surface  of  the  globe,  one 
being  the  place  of  deiiarture  and  the  other  that  of  destination.  Let  P 
represent  the  pole,  PA  and  PB  the  meridians  passing  through  A  and 
B.  Let  a  great  circle  pass  through  A  and  B.  The  points  a,  v,  h  are 
on  the  equator.  From  P  draw  PV  perpendicular  to  the  great  circle, 
then  Y  will  be  the  vertex. 

In  the  spherical  triangle  APV, 

tan(90-Lv)       cot  L^  ,,> 

cos  APv  =.        ,n,\       T  N  or  ^  ■  T-  (1) 

tan  (00  —  L|)        cot  ui  ^  ' 

12 


13 

111  the  spl)eri»-al  triiiiijulo  I>PV, 


cos  BPV  =       .  j"  (2) 

cot  L  ^   ' 


cot  Lv 
U 

These  two  eciuations  remain  the  same,  whetlier  the  perpeiulicular 
falls  within  or  withont  the  triangle,  for  all  possible  values  of  the  parts 
named.  In  either  eiiuation,  any  two  of  the  terms  being  given,  the  third 
becomes  known;  and  from  the  similarity  of  (1)  and  (2)  it  is  obvious  that 
each  of  these  equations  comi)uted  for  all  possible  values  of  Lv  and  Li 
or  L^  will  give  the  same  series  of  results  for  the  arc  APV  or  BP\',  and 
embraces  all  the  values  that  the  arc  can  have. 

The  successive  values  of  this  arc  were  computed  for  all  the  integral 
values  of  Lv  and  Li  from  1°  to  89°,  inclusive.  The  results  form  the  dis- 
tance column  of  Towson's  tables.  They  were  also  projected  as  ordiiiates 
to  the  axes  named  Meridians  of  Vertex  in  the  linear  index,  which  is 
appended,  and  the  curves  were  drawn  through  the  extremities  of  these 
ordiiiates. 

Supjiose  the  vertex  of  the  great  circle  passing  through  two  places, 
A  and  ii,  is  to  be  found;  with  a  pair  of  dividers  take  out  their  differ- 
ence of  longitude  from  the  scale  of  differences  of  longitude,  and,  if  the 
latitudes  of  A  and  B  have  the  same  name,  set  one  point  of  the  dividers 
in  the  division  of  the  diagram  nearest  to  the  right  hand  on  that  curve 
whose  number  corresponds  to  the  latitude  of  either  A  or  B,  whichever 
is  the  nearer  to  theecjuator,  and  keeping  the  line  joining  the  points  of 
the  dividers  horizontal,  follow  the  curve  up  or  down  till  the  other  point 
meets  the  curve  corresponding  to  the  latitude  of  the  other  place.  The 
index  linepassing  through  the  points  of  the  dividers  in  this  position  will 
indicate  the  latitude  of  the  vertex.  If  that  point  of  the  dividers  which 
stands  on  the  curve  corresponding  to  the  latitude  of  departure  be  kept 
fixed,  and  the  other  be  moved  in  a  horizontal  direction  until  the  nearest 
meridian  of  vertex  is  reached,  the  dividers,  being  applied  to  the  scale, 
will  indicate  the  longitude  of  the  point  of  departure  from  the  vertex. 
With  the  latitude  of  the  vertex  and  longitude  from  the  vertex,  thus 
found,  the  tables  are  entered  and  the  distance  from  the  vertex  and  the 
course  are  picked  out.  If  the  latitudes  of  A  and  B  are  of  different 
names,  the  vertical  line  named  Equator  in  the  linear  index  must  be  kept 
between  the  points  of  the  dividers  in  finding  the  latitude  of  the  vertex, 
but  the  rest  of  the  operation  remains  the  same  as  before  described. 

DEICHMAN'S  METHOD. 

In  1857  Mr.  A.  H.  Deicuman,  in  full  knowledge  of  and  with  a  view 
to  improving  what  had  already  been  done  by  Towson,  devised  a  dia- 
gram called  a  Scale  of  Great  Circles  to  be  used  in  connection  with  a 
table  for  finding  great  circle  courses  and  distances.  The  table  is  the 
same  as  Towson's,  except  that  the  longitudes  and  distances  are  reckoned 
from  the  intersection  of  the  great  circle  with  the  equator,  instead  of  the 
vertex.  The  Scale  of  Great  Circles  is  a  device  for  finding  the  vertex  of 
any  great  circle,  which  is  all  that  is  necessary,  for  when  the  vertex  of  a 
great  circle  is  known  all  courses  and  distances  on  that  great  circle  are 
virtually  known.    Deichman's  diagram  or  Scale  of  Great  Circles  con- 


14 

sists  of  a  Mercator  projection,  of  such  scale  as  to  make  tbe  representa- 
tion of  oiie-fourtli  of  the  earth's  surface  of  convenient  size,  with  a  series 
of  great  circles  projected  upon  it. 

His  method  of  finding  the  vertex  consists  in  cutting  out  of  paper  or 
card-board,  on  the  same  scale  as  that  of  the  Scale  of  Great  Circles,  an 
"  Index  Model,"  whose  two  upper  edges  shall  represent  the  positionsof 
the  two  places  through  which  the  great  circle  jiasses  in  their  approxi- 
mate latitudes  and  longitudes,  and  sliding  this  (see  figure)  over  the 
Scale  of  Great  Circles  until  both  the  upper  edges  shall  lie  on  the  same 
great  circle.  The  latitude  of  the  vertex  is  then  read  off  from  the  Scale 
of  Great  Circles. 


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brevoort's  method. 

Mr.  J.  Carson  Brevoort  submitted  to  the  Hydrographic  OfiBce,  for  an 
opinion,  in  July,  1887,  a  method  for  facilitating  great  circle  sailing 
which  closely  resembles  the  methods  of  Towson  and  Deichman.  His 
tables  are  the  same  as  Towson's,  except  that  the  column  of  distances  is 
omitted  and  a  column  of  compass  courses,  reading  to  one-eighth  of  a 


90°  80'  70°    60°  50°  40°  30°  20°    10°     0°     10°  20°  30°  40°   SO'  60°  70°  S0°  90_ 


point,  is  added.      The  index  diagram  is  a  Mercator  projection,  laid 
down  ou  transparent  material,  with  a  system  of  great  circles  projected 


15 

upon  it,  like  Beiclinian's  Scale  of  Great  Circles.  This  is  designed  to  slide 
over  a  Mercutor  chart  on  the  same  scale  as  that  of  the  index  diagram, 
with  the  equators  of  the  diagram  and  the  chart  in  coincidence,  until  the 
two  places  between  wliicli  tlie  great  circle  track  passes  lie  on  the  same 
great  circle  of  the  diagram.  Jn  the  figure  the  lines  rei)resented  by 
dashes  are  those  which  occur  on  the  transparent  diagram.  It  will  thus 
be  seen  that,  while  Deichman's  method  causes  any  part  of  the  earth  to 
revolve  over  one  system  of  great  circles,  Brevoort's  causes  one  system 
of  great  circles  to  revolve  over  any  part  of  the  earth. 

BERGEN'S  METHOD. 

In  1857  W.  C.  Bergen,  master  in  the  British  mercantile  marine,  pub- 
lished a  set  of  spherical  tables  of  general  adaptability  to  the  i)roblems 
of  nauti(!al  astronomy,  and  a  diagram,  by  the  aid  of  which  the  results 
ill  the  tables  are  rendered  ap]>licable  to  the  problem  of  great  circle 
sailing. 

Although  this  method  can  not  be  used  with  the  same  degree  of  facil- 
ity as  some  of  the  others  of  the  same  character,  it  is  well  worth  atten- 
tion on  account  of  the  radical  principles  involved. 

The  diagram,  which  is  appended,  represents  a  quadrant  of  the  earth's 
surface  with  the  parallels  of  latitude  and  the  meridians  of  longitude 
from  degree  to  degree.  Each  fifth  parallel  and  fifth  meridian  is  num- 
bered and  drawn  stronger  than  the  others,  and  in  order  further  to  distin- 
guish thein  the  alternate  parallels  between  them  are  dotted.  Over  this 
net- work  of  parallels  and  meridians  a  system  of  great  circles  is  drawn. 

Under  the  equator  the  degrees  from  the  meridian  of  the  vertex  are 
numbered, thus  showing  the  longitude  from  the  vertex,  and  the  half  differ- 
ence of  longitude;  and  under  this  again  these  numbers  are  doubled, 
indicating  the  difference  of  longitude;  and,  to  obviate  the  necessity  of 
taking  the  supplement  of  the  difference  of  longitude,  in  the  case  of  the 
places  being  on  opposite  sides  of  the  equator,  the  third  line  of  numbers 
is  given,  containing  the  difference  of  longitude  reckoned  from  the  point 
where  the  great  circle  crosses  the  eqnator. 

The  object  of  the  diagram  is  to  find  the  latitude  of  the  vertex  of  the 
particular  great  circle  which  passes  through  the  points  of  departure 
and  destination,  and  the  longitude  of  the  point  of  departure  from  the 
meridian  passing  through  the  vertex. 

In  the  figure,  let  C,  the  center  of  the  primitive  PELR,  be  the  projec- 
tion of  a  point  on  the  earth's  equator,  PL  that  of  a  meridian  i^assing 
through  the  point  0,  and  ER  that  of  the  equator.  Let  P  represent 
the  nearer  pole,  and  Vmh  and  Pnh  meridian  circles  equally  inclined 
to  the  primitive.  Then,  it  is  evident  that  PL  bisects  the  angle  mPn, 
and  therefore  viFn  is  double  the  angle  CPw. 

Again,  let  a  great  circle  EAR  be  drawn  through  EU,  cutting  Pwj, 
PL,  and  Vn  in  points  y,  A,  z.  And  through  yz  let  the  parallel  of 
latitude  ^y^i'^o  be  drawn. 


16 

Now,  if  the  quadraut  PCE  be  couceived  to  turn  around  the  radius 
PC,  wliicb  remains  fixed,  and  to  be  placed  on  the  quadrant  PCK,  the 
projections  of  the  one  will  fall  upon  the  corresponding  projections  of  the 
other,  and  will  coincide  with  them;  Pm  will  therefore  coincide  with 
Pn,  AE  with  AE,  p.v  with  ox,  and  the  point  ij  with  the  point  s. 


A  is  the  vertex  of  the  great  circle  EAR,  PL  is  the  meridian  of  the 
vertex,  and  CP»  is  the  longitude  of  the  point  z  from  the  vertex. 

Hence,  when  two  places  are  given  on  the  same  parallel  of  latitude, 
to  find  the  latitude  of  the  vertex  and  the  longitude  from  the  vertex,  let 
Cw  be  equal  to  half  the  difference  of  longitude,  and  ox  represent  part  of 
the  parallel  of  latitude  passing  through  the  places ;  with  a  pointed  in- 
strument trace  up  the  meridian  P??,  and  with  another  trace  along  the 
parallel  ox  until  the  curves  meet  in  the  point  z,  then  trace  the  great 
circle  K;jAE,  passing  through  z,  until  it  meets  the  meridian  of  vertex  in 
the  point  A.  Then  CA  is  the  latitude  of  the  vertex,  and  Cn  is  the  lon- 
gitude from  the  vertex. 

Again,  if  the  places  are  on  the  same  side  of  the  equator  and  ou  differ- 
ent parallels  of  latitude,  ox  and  rq,  let  Pn,  Ps,  and  P^  represent  mericV- 
ians  equidistant  from  one  another,  and  let  Cs  be  equal  to  half  the  dif- 
ference of  longitude.  With  the  right  hand  trace  up  the  meridian  I's 
until  it  cuts  the  lower  parallel  of  latitude,  rq,  then,  with  the  left  hand, 
trace  it  up  again  until  it  meets  the  higher  parallel  of  latitude  ox,  then 
trace  the  lower  latitude  to  the  right  hand  and  the  higher  to  the  left, 
being  careful  to  move  an  equal  number  of  degrees  on  each  side  of  the 
meridian  P.s,  until  the  curve  of  the  great  circle  passes  through  both 
points.  Let  these  points  be  represented  by  z  and  u,  and  let  livzA  rep- 
resent the  great  circle,  then  AC  will  represent  the  latitude  of  the  ver- 
tex, and  C?i  and  (M  the  longitudes  of  the  points  z  aud  u  from  the  ver- 
tex. 

It  is  evident  that  \(  y  be  the  place  of  the  ship  and  u  that  of  her  des- 
tination, or  conversely,  the  meridian  of  vertex  falls  between  them,  ana' 


17 


'POLE       t) 


c 


17 

that  win  added  to  nt  is  the  diflEereuce  of  longitude;  but  C?t  is  equal  to 
hiilt'  o(  inn,  and  therefore  twice  Cn  added  to  nt  is  equal  to  the  difference 
of  longitude.  IS^ow,  ns  is  half  of  nt,  and  when  twice  C.s  is  considered  as 
the  difference  of  longitude,  when  the  hand  is  moved  to  the  left  from  s 
to  n,  twice  ns  is  taken  from  the  difference  of  longitude,  but  the  other 
hand  being  moved  from  s  to  f,  they  are  distant  from  each  other  nt,  which 
is  equal  to  twice  ns,  and  7it  being  added  to  twice  Cn,  the  difference  of 
longitude  is  the  same  as  at  first. 

If  the  meridian  of  the  vertex  fall  outside  of  the  portion  of  the  great 
circle  track  which  lies  between  the  points  of  departure  and  destination, 
as  in  the  case  in  which  z  and  u  are  taken  to  represent  those  points,  let 
CH  be  equal  to  ns,  which  is  equal  to  the  half  difference  of  longitude 
between  z  and  n,  and  let  Pfl  be  traced  uj)  as  before.  In  this  case  the 
left  hand  will  meet  the  meridian  of  vertex  without  meeting  the  great 
circle,  and  the  right  hand  must  be  moved  until  it  is  at  a  distance  from 
the  meridian  of  the  vertex  equal  to  the  difference  of  longitude.  Both 
hands  must  then  be  moved  equally  until  they  arrive  at  the  points  z  and 
u  where  the  great  circle  passes  through  them. 

For  the  explanation  of  the  case  in  which  the  points  of  departure  and  des- 
tination are  on  opposite  sides  of  the  equator,  imagine  the  opposite  pole 
of  the  primitive  to  be  taken  as  the  projecting  i)oiut,  and  equal  circles 
to  be  projected  upon  the  lower  right  hand  quadrant  LCR,  and  con- 
ceive it  to  be  turned  upon  the  radius  CR  as  an  axis  until  it  shall  coin- 
cide with  the  quadrant  PCR.  The  projections  of  one  will  then  coincide 
with  the  corresponding  projections  of  the  other  ;  and  if  the  places  are 
in  equal  latitudes,  as  z  and  ^',  then  hR  will  be  half  the  difference  of  long- 
itude, and'C«  will  be  half  its  supplement.  Therefore  by  taking  the  sup- 
plement of  the  difference  of  longitude  and  proceeding  exactly  as  in 
the  other  cases  the  latitude  of  the  vertex  AC  or  A'C  and  the  longitude 
of  the  vertex  Cn  or  C^  will  be  found. 

The  latitude  of  the  vertex  and  the  longitude  from  the  vertex  having 
been  found  from  the  diagram  by  inspection,  a  table,  of  which  the  fol- 
lowing is  a  specimen  i)age  heading,  is  entered,  and  the  course  and  the 
distance  from  the  vertex  are  picked  out. 

Latitude  of  vertex,  19°. 


Longittido 
from  vertex. 

Distance 
from  vertex. 

Course. 

1° 

0°  ",' 

0O19' 

In  the  actual  construction  of  the  tables  the  author  greatly  abridged 
his  work  by  observing  the  trigonometric  principle  that,  if  the  comple- 
ments of  the  hypotenuse  and  base  of  a  right-angled   spherical  tri- 
19862 2 


18 

angle  be  taken  as  the  base  and  hypotenuse  of  another  to  the  same 
angle,  the  perpendicular  of  each  triangle  is  equal  to  the  complement  of 
the  arc  which  measures  the  remaining  angle  of  the  other  triangle. 

Let  the  figure  represent  a  projection  on  the  plane  of  the  primitive. 
Let  P  and  P'  be  the  poles,  EQ  the  equator,  EVQV  any  great  circle, 
whose  northern  and  sonthern  vertices  are  at  V  and  V,  respectively,  and 
PLMP'  the  meridian  passing  through  the  i»oint  L  on  the  great  circle 
EVQV,  at  which  the  course  and  distance  are  required. 


The  angle  LQM,  between  the  plane  of  tlje  equator  and  the  plane  of 
the  great  circle,  is  numerically  equal  to  the  arc  VO,  or  the  latitude  of 
the  vertex  V.  In  the  triangle  LMQ,  the  side  LQ  is  the  complement 
of  the  distance  of  the  point  L  from  the  vertex,  the  side  MQ  is  the  com- 
plement of  the  longitude  of  L  from  the  vertex,  the  sideLM  is  a  measure 
of  the  latitude  of  L,  the  angle  MLQ  represents  the  course,  and  the  angle 
LQM  the  latitude  of  the  vertex.  In  the  construction  of  the  tables,  the 
triangle  LMQ  was  computed  to  every  degree  of  the  angle  Q,  and  also  of 
the  arc  LQ,  until  LV  was  equal  to  or  next  less  than  MQ.  Then  the  com- 
plements of  LQ  and  MQ  were  taken,  and  the  trigonometric  principle 
already  mentioned  was  applied. 

With  regard  to  the  degree  of  dependence  to  be  placed  upon  this 
method,  Mr.  Bergen  remarks,  in  his  writings  on  the  subject,  that,  by 
means  of  an  extensive  induction,  he  arrived  at  the  conclusion  that  for 
a  diflerence  of  longitude  equal  to  about  15^,  the  course  may  be  de^ 
pended  upon  generally  to  less  than  one-eighth  i)oint;  in  some  instances, 
■when  the  diflerence  of  longitude  is  small  and  the  latitude  high,  the 
course  may  be  erroneous  nearly  one-quarter  of  a  point ;  but  this  will  not 
happen  with  a  difference  of  longitude  more  than  15°  and  a  latitude  less 
than  600. 


19 


THE    TERRESTRIAL    GLOBE. 


The  Strict  practical  solution  of  the  problem  of  great  circle  sailing, 
which  consists  in  using  a  terrestrial  globe,  is  of  ecpuil  convenience  and 
of  at  least  equal  accuracy  with  the  foregoing  niethotl.  A  great  circle 
passing  through  any  two  places  upon  the  earth's  surface  may  be  traced 
upon  a  terrestrial  globe  by  elevating  or  depressing  the  i)olar  axis  and, 
at  the  same  time,  turning  the  globe  until  the  places  coincide  with  the 
upper  edge  of  the  wooden  horizon.  In  this  position  of  the  globe  the 
upper  edge  of  the  wooden  hori/on  represents  the  great  circle,  and  can 
be  used  as  a  ruler  for  tracing  it  upon  the  surface  of  the  globe. 

Having  thus  traced  upon  the  globe  the  great  circle  passing  through 
the  two  given  jtlaces,  their  distance  apart  is  shown  by  tlio  number  of 
degrees  intercepted  between  them  on  the  Avooden  horizon,  the  latitude 
and  longitude  of  the  vertex  can  be  read  oft',  and  the  various  places 
tlirough  which  the  great  circle  passes  are  shown. 

It  will  thus  be  seen  that,  instead  of  tinding  the  latitude  of  the  vertex 
and  longitude  from  the  vertex  by  the  diagrams  which  have  been  de- 
vised by  Towson,  Deichman,  Bergen,  and  Brevoort,  a  terrestrial  globe 
can  be  used  for  finding  these  elements  with  quite  as  nnich  efficiency. 
After  which  the  spherical  tables,  which  have  been  arranged  by  any  of 
these  authors  can  be  used  for  finding  the  courses  and  distances. 

Or  the  latitudes  and  longitudes  of  a  sufiicient  number  of  points  or 
places  on  the  track  of  a  great  circle,  thus  traced  upon  a  terrestrial  globe, 
n)ay  be  read  oif  and  the  corresponding  positions  plotted  on  a  Mercator 
sailing  chart.  Then,  connecting  the  points  by  drawing  a  curved  line 
through  them,  a  graphical  representation  of  a  great  circle  upon  the 
chart  is  obtained  without  any  computation. 

THE   DIRECT   TRACK  SCALE. 

This  was  recently  designed  by  Mr.  Gustave  Herrle,  of  the  U.  S.  Hy- 
drogTai)hic  OfiBce,  as  an  appendage  to  the  Mercator  chart,  with  a  view 
of  making  the  chart  more  directly  available  in  pursuing  a  great-circle 
track.  It  consists  of  a  gnomonic  projection,  with  the  point  of  contact 
on  the  equator,  contracted  in  longitude.  From  this  the  position  of  any 
point  of  a  great-circle  track  can  be  read  oft"  and  transferred  to  the 
Mercator  chart.  There  are  also  logarithmic  scales  of  sines  and  cosines 
for  the  measurement  of  courses  and  distances. 

The  gnomonic  projection,  with  the  point  of  contact  on  the  equator,  is 
that  which  is  used  by  Lieutenant  Ililleret,  French  navy,  whose  great 
circle  sailing  charts  will  be  referred  to. 
Let  r  represent  the  radius  of  the  sphere; 

f^  represent  the  longitude  of  any  meridian   reckoned  from  the 
meridian  passing  through  the  point  of  contact,  or,  as  it  is  gen- 
erally called,  the  middle  meridian; 
cp  represent  the  latitude  of  any  parallel. 
Then  the  successive  distances  of  the  parallel  straight  lines  of  the  pro- 


20 


jection,  which  represent  the  meridians,  from  the  middle  meridian,  will 
be  r .  tan  6;  and  the  points  at  which  the  successive  parallels  of  latitude 
cut  the  middle  meridian  will  be  r  .  tan  (p,  measured  from  the  equator. 

The  distance  above  the  equator  at  which  any  i^arallel  whose  latitude 
is  (p  cuts  a  meridian  whose  longitude  from  the  middle  meridian  is  6  will 
be  r  .  tan  cp .  sec  6. 

By  the  application  of  the  preceding  expressions  the  gnomonic  equa- 
torial projection  of  the  Northern  Hemisphere  up  to  latitude  G0°,  shown 
in  the  above  figure,  has  been  constructed. 

In  order  to  economize  area  and  yet  preserve  all  the  properties  of  the 
projection  in  relation  to  direct  tracks  the  first  projection  is  projected 
orthogra|)hicany  upon  a  second  plane,  which  j)ayses  through  a  me- 
ridiiin  of  the  first  plane  of  projection  aud  makes  with  it  any  angle,  a. 

The  degree  of  contraction  will  then  be  directly  proportional  to  the 
cosine  of  the  angle  a,  so  that  the  contracted  projection  could  be  con- 
structed at  once  by  inserting  in  the  expression  for  distances  of  the  suc- 
cessive meridians  from  the  middle  meridian  the  cosine  of  the  angle  of 
inclination  of  the  two  i)rojecting  planes.  This  expression  would  there- 
fore be  r  .  tan  6 .  cos  a. 

The  expression  for  constructing  the  parallels  of  latitude  will  remain 
the  same,  i.  e.,  r  .  tan  ^  sec  0. 

THE    MEASUREMENT    OF    COURSES  AND  DISTANCES  ON    THE    DIRECT 

TRACK    SCALE. 


In  the  figure,  let  AB  represent  any  great  circle  of  the  earth,  A  the 
place  of  departure  or  the  ship's  position  at  any  time,  and  B  the  place 
of  destination. 
Let  C  represent  the  course  at  A, 

Y  represent  the  vertex  of  the  great  circle  AVB, 
d  represent  the  distance  between  A  and  B, 
Lg  represent  the  latitude  of  A, 
Lv  represent  the  latitude  of  V, 
Lp  represent  the  latitude  of  B, 
A,  represent  the  difference  of  longitude  between  A  and  B. 


COURSE   a    DISTANCE   SCALES. 


dJnnjJTTbiffliiirWi.iffliJtm^y^ji'liffl^laatsq^ 


,  Distance  Scale; 


r 


21 
From  the  right-angled  spherical  triangle  APV, 

siu  C  :  sin  90°  ::  cos  Lv  :  cos  Lg 

sin  C=«£iilr 
cos  Lg 

log  sin  C=log  cos  L,.— log  cos  Lg  (1) 

From  the  spherical  triangle  APB, 

sin  C  :  sin  A  ::  cos  L,,  :  sin  d 

log  sin  (?=log  sin  A+log  cos  L^— log  sin  C.  (2) 

In  order  to  solve  the  logarithmic  equations  (1)  and  (li)  graphically, 
and  thns  find  the  coarse  at  any  point  and  the  distance  from  that  point 
to  the  place  of  destination  without  computation,  a  length  is  adopted 
corresponding  to  a  certain  number  of  units  of  the  mantissa  of  a  log 
sin  or  log  cos,  and  a  linear  scale  is  thus  constructed  giving  the  necessary 
angles. 

The  Direct  Track  Scale  and  its  accompanying  logarithmic  scales  are 
here  shown  reduced  to  one-half  the  size  which  it  is  proposed  to  use  in 
practice. 

The  following  rules  for  using  this  method  are  given  by  Mr.  Herrle: 

On  the  meridian  of  the  diagram  marked  0",  plot  the  latitude  of  the 
place  of  departure,  and  on  the  meridian  whose  longitude  from  the  me- 
ridian marked  0°,  is  equal  to  the  difference  of  longitude  between  the 
places  of  departure  and  destination,  plot  the  latitude  of  the  place  of 
destination,  connect  the  two  points  thus  plotted  by  a  straight  line  and 
find  by  inspection  on  the  diagram  the  highest  latitude  which  it  reaches. 
This  latitude  will  be  L,.,  the  latitude  of  the  vertex  of  the  great  circle 
which  passes  through  the  places  of  departure  and  destination. 

Find  Lg  and  L^  by  the  lower  readings  on  the  logarithmic  scale ;  take 
their  difference  and  lay  it  off  irom  the  right  end  of  the  scale  toward  the 
left.     The  upper  reading  of  the  point  laid  off  corresponds  to  C. 

Courses  greater  than  28°  may  be  measured  on  either  scale,  courses 
less  than  28°  on  the  lower  scale  only. 

For  the  measuremeut  of  distances,  find  C  at  L^  and  A  on  the  upper 
readings  and  L,,  on  the  lower  readings ;  take  the  difference  between  C 
and  A  and  lay  it  off  from  Lp  toward  the  right  or  left,  according  as  the 
place  of  destination  is  to  the  right  or  left  of  the  ship's  position.  The 
upper  reading  of  tlie  point  laid  oft'  corresponds  to  d,  the  distance. 

Distances  less  than  28°  or  1,G80  nautical  miles  can  be  measured  on  the 
lower  scale  only. 


SECTION  II. 


METHODS  J)EPENDmG  UPON  THE  (INOMONIC  CHART. 

The  object  of  cLiirts  is  to  exhibit,  by  suitable  representation,  on  a 
reduced  scale  and  on  a  plane  surface,  the  relative  positions  of  points, 
lines,  or  objects  on  the  earth's  surface;  and  since  such  positions  are 
usually  defined  by  spherical  coordinates,  the  primary  object  of  the  pro- 
iection  upon  which  the  chart  is  based  is  the  delineation  of  these  circles 
of  reference  according  to  certain  assumed  or  fixed  geometrical  laws. 
Any  point,  line,  or  object  intended  for  representation  may  then  be  laid 
down  by  means  of  its  known  coordinates,  and,  conversely,  the  coordi- 
nates of  any  plotted  point  may  be  ascertained.  The  gnomonic  chart  is 
based  upon  a  system  of  projection  in  which  the  jilaue  of  projection  is 
tangent  to  tlie  surface  of  the  sphere,  and  the  eye  of  the  spectator  is 
supposed  to  be  situated  at  the  center  of  the  sphere,  where,  being  at 
once  in  the  plane  of  every  great  circle,  it  will  see  tliese  circles  i^rojected 
as  straight  lines  where  the  visual  rays,  passing  through  them,  intersect 
the  plane  of  projection. 

godfray's  cinomonic  chart. 

In  1858  Hugh  Godfray,  M.  A.,  Fellow  of  the  Cambridge  Thil.  Soc, 
prepared,  for  the  jjurpose  of  great  circle  sailing,  two  gnomonic  charts 
covering  the  greater  part  of  the  world — one  for  the  Northern  and  one 
for  the  Southern  Hemisphere.  He  used  as  points  of  contact  the  geo- 
graphical poles  of  the  earth. 

The  i)arallels  of  latitude  are  tlierefore  represented  by  a  series  of  con- 
centric circles  wiiose  radii  are  equal  to  r  tan  (90°  — hit.)  or  r  cot  lat., 
where  r  is  any  convenient  linear  magnitude,  and  represents  the  radius 
of  the  sphere.  The  meridians  are  straight  lines  drawn  from  the  com- 
mon center  or  pole,  dividing  each  circumference  into  three  hundred  and 
sixty  (Hjual  parts.  Any  one  of  these  being  selected  for  the  prime 
meridian,  the  coast  lines  of  different  countries  may  then  be  traced  in 
the  usual  manner  by  means  of  the  longitudes  from  the  i^rime  meridian 
22 


GREAT  CIHCLE  SAIUNd 

lOITnSKSSDISTAMES 


23 

and  the  latitudes  of  the  diiferent  i)oints.  Mr.  Godfray  also  devised 
means  tending  toward  convenience  for  the  measurement  of  courses  and 
distances  on  the  grc^at  circle  track.  His  diagram  fur  tliis  jmrimse,  con- 
structed upon  a  separate  slieet,  measuring  1")  by  19  inches,  consists  of 
a  series  of  concentric  curves  corresponding  to  the  ])ara]lels  of  latitude, 
bounded  by  a  vertical  line,  AC,  and  a  horizontal  line,  AB.  The  hori- 
zontal line  is  divided  into  seventy  equal  parts,  representing  the  degrees 
of  latitude  from  20°  to  90^. 

The  vertical  line  AC  is  a  scale  of  distances  from  the  highest  lati- 
tude and  is  divided  into  twenty-one  equal  parts,  each  representing  200 
nautical  miles.  Through  the  various  points  of  division  of  these  scales 
are  drawn  horizontal  and  vertical  straight  lines  over  tlie  whole  diagram. 
The  concentric  curves  corresponding  to  the  parallels  of  latitude  are 
traced  from  the  following  considerations: 
Let  ABC  represent  any  great  circle, 

B  represent  its  vertex, 

P  represent  the  pole, 

Lv  represent  the  latitude  of  the  vertex, 

Li  represent  any  lower  latitude  on  the  great  circle, 

<1  represent  the  distance  from  the  vertex, 
we  have  from  the  right  angled  spherical  triangle  L'BC,  by  Napier's 
analogies  : 


cos  (90^^— L,)=cos  d  cos  (90°— L,.) 

cos  <Z=cos  (90°— Li)  sec  (90°— Lv)=sin  Li  cosec  Ta 

which  determines  the  distance  d  at  which  the  latitude  curve  L,  crosses 
the  horizontal  line  passing  through  the  division  L,. 

Thirty-two  curved  lines  are  drawn  radially  from  the  point  nuirking 
90"^  latitude  and  zero  distance.  The  si)aces  between  these  are  alter- 
nately light  and  shaded,  and  are  marked  in  points  and  (juarter-])oints 
for  the  determination  of  courses. 

At  the  vertex  of  a  great  circle  the  course  is  due  east  or  west,  and 
as  we  move  along  the  great  circle  away  from  the  vertex  the  course  will 
alter  continuously  as  the  distance  alters.  The  connection  between  the 
course  and  distance  will  be  determined  by  reference  to  the  right-angled 
spherical  tiiangle  in  which  ^  represents  the  course  at  any  point  whose 
distance  is  d  from  the  vertex.  From  this  triangle,  sin  rf=cot  ^  cot  Lv, 
which  determines  the  distance  of  the  point  where  the  course  is  f;  and 
the  curve  for  the  course  f^iu  the  diagram  must  pass  through  that  jjoint 


24 

where  the  vertical  line  correspoudiug  to  the  highest  latitude  L^  is  met 
by  the  horizontal  line  at  the  distance  d. 

CfODFKAY'S   BOURSE   AND   DISTANCE   DIAGRAM. 


The  following  example  will  illustrate  the  use  of  the  diagram : 
Suppose  it  is  required  to  find  the  successive  courses  and  distances  to 


25 

be  run  on  each  course  between  two  places,  A  and  B.  Draw  a  straight 
line  connecting;"  A  and  B  on  the  gnomonic  chart,  and  observe  the  highest 
latitude  reached  by  tliislinc  between  A  and  B,  or  ])roduced.  8ui)pose 
the  highest  latitude  is  Ibiind  to  be  ik-V^  S.  and  the  latitude  of  A,  the 
point  of  departure,  is  35°  S.  Now,  refer  to  the  diagram  and  find  the 
point  where  the  vertical  line  through  44.]^  is  crossed  by  the  'S~i^  curve. 
This  falls  on  the  light  spa(;e  corresponding  to  G  jjoints.  Hence  the  first 
course  is  S.  67^°  E.  or  W.,  according  as  B  is  east  or  west  of  A.  Then, 
pro<;eeding  along  the  vertical  line  44i°  toward  the  highest  hititude, 
measure  the  breadths  of  the  successive  light  and  shaded  si)a('e8,  and 
the  recpiired  series  of  courses  and  distances  will  be  obtained. 

knork's  gnomonio  chart. 

In  1860  Mr.  E.  R.  Knorr,  of  the  U.  S.  Hydrographi('  Office,  designed  a 
chart  of  the  North  Atlantic  ocean  on  the  guomonic  projection,  with  a 
view  to  exhibiting  at  once  all  the  influences  controlling  the  navigation 
of  that  ocean.  To  this  end  his  chart  embraces  the  hydrograjjhy  of  the 
ocean,  the  top<)grai)hy  of  its  shores  aiul  islands,  the  forces  and  directions 
of  the  winds  which  may  be  expected  at  different  seasons,  the  general  flow 
of  the  currents,  and  all  great  circle  arcs,  with  means  for  finding  courses 
upon  them.  Tiie  plane  of  projection  is  tangent  to  the  sphere  in  latitude 
30°  north  and  longitude  40^  west.  The  meridians  of  longitude  are  there- 
fore represented  by  a  series  of  straight  lines  converging  toward  the 
north,  and  the  parallels  of  latitude  by  a  series  of  elliptic  and  hyperbolic 
curves.  The  shortest  distance  between  any  two  points  on  this  chart  is 
the  distance  on  a  straight  line  Joining  them,  since,  according  to  the 
princiijles  of  the  gnomonic  projection,  every  straight  line  represents  a 
portion  of  the  circumference  of  a  great  circle.  For  the  measurement  of 
coiu-ses  a  contracted  form  of  Mr.  John  T.  Towsou's  tables  is  i)repared 
upon  the  face  of  the  chart.  These  tables  are  fully  treated  of  in  the  ref- 
erence to  Towson's  method.  The  latitude  of  the  vertex  of  the  great  circle 
track,  which  is  recpiired  in  finding  the  courses  from  these  tables,  is  readily 
found  by  inspection  on  the  chart  in  all  ordinary  cases.  For  those  cases 
in  which  the  vertex  of  the  great  circle  falls  beyond  the  upper  border  of 
the  chart  simple  means  of  considerable  convenience  are  given  for  find- 
ing the  vertex.  The  author  gives  a  rule  for  computing  great  circle  dis- 
tances, and  states  in  a  note  on  the  chart  that  distances  can  not  be 
directly  measured  on  the  gnomonic  projection.  This  chart  was  never 
publishe<l,  because  in  its  chief  feature,  that  of  facilitating  great  circle 
sailing,  it  was  (piickly  supplanted  by  better  methods. 

hilleret's  gnomonic  chart. 

In  1879  Lieutenant  Uilleret,  French  navy,  designed,  for  the  purpose 
of  great  circle  sailing,  a  set  of  gnomonic  charts,  with  the  point  of  con- 


26 

tact  ou  the  equator.  The  parallels  of  latitude  are  therefore  represented 
by  a  series  of  hyperbolas  whose  transverse  axes  are  e(|ual  to  r.  tau.  lat- 
itude, and  whose  conjugate  axes  are  eijual  to  r,  where  r  is  any  conven- 
ient linear  magnitude  and  re])resents  the  radius  of  the  sphere.  The 
meridians  are  parallel  straight  lines  whose  distances  from  the  meridian 
passing  through  the  point  of  contact  are  equal  to  r.  tau.  longitude, 
reckoned  from  the  point  of  contact.  For  the  measurement  of  courses 
there  is  added  a  Mercator  projection  covering  10°  of  longitude,  and 
of  the  same  extent  in  latitude  as  the  northern  or  southern  portion 
of  the  chart.  The  great  circle  track  being  drawn  upon  the  chart,  the 
latitude  of  the  point  of  departure  and  that  of  a  })oint  two  or  three  de- 
grees of  longitude  from  the  point  of  departure  toward  the  point  of  des- 
tination are  read  from  the  chart.  These  points  are  then  idotted  on  the 
Mercator  projection  in  their  respective  latitudes  and  with  their  correct 
difference  of  longitude,  and  the  bearing  of  the  line  joining  them,  as 
measured  by  means  of  a  protractor,  is  read  oft"  for  the  first  course.  The 
succeeding  courses  are  determined  in  the  same  manner.  This  method 
is  approximate.  The  direct  means  for  the  measurement  of  distances 
exist  in  the  projection.  Since  in  this  projection  all  great  circles  which 
are  equally  distant  from  the  point  of  contact  must  each  have  an  angu- 
lar unit  represented  by  the  same  linear  magnitude,  and  since  the  merid- 
ians are  great  circles,  to  determine  the  length  of  any  great  circle  track 
we  have  simply  to  let  fall  a  perpend i(;ular  from  the  point  of  contact 
to  the  track,  lay  off  this  perpendicular  distance  from  the  point  of  con- 
tact along  the  equator,  and  draw  a  meridian  through  the  extremity  of 
it.  The  i)ortion  of  the  great  circle  track  on  one  side  of  the  foot  of  the 
perpendicular  being  laid  off  above  the  ecjuator,  and  that  on  the  other 
side  below  the  equator,  the  difference  of  latitude  between  tlie  two  points 
thus  marked  will  be  the  required  distance. 

jenzen's  gnomonic  chart. 

Mr.  Oarl  Jenzeu,  master  mariner,  presented  for  examination  at  the 
Hydrogra])hic  Oftice,  Navy  Department,  in  July,  18S7,  a  gnomonic  chart 
of  the  North  Atlantic  Ocean,  which  combined,  though  without  his 
knowledge,  the  method  of  i)rojection  used  by  Mr.  Godfray  in  1858  with 
the  method  of  measuring  (courses  which  was  ado[)ted  and  published  by 
the  French  in  1879.  There  exists  a  slight  difference  between  the  course 
diagrams  of  Lieutenant  Hilleret  and  Mr.  Jenzen.  The  former  used  a 
pure  Mercator  projection  while  the  latter  preserves  an  equality  in  the 
degrees  of  latitude  throughout  his  diagram  and  causes  the  meridians 
to  converge  in  the  ratio  of  the  cosine  of  the  latitude.  Both  methods 
are  approximate.  Of  all  the  available  points  of  contact  on  the  sphere, 
considered  with  reference  to  their  adaptation  to  modern  wants  in  great 
circle  sailing,  tlie  geograpliical  i>ole  is  least  a(la})ted.  No  complete 
gnomonic  chart  of  the  Indian  Ocean  could  be  made  on  this  i)lan,  no 


27 

means  could  be  devised  by  which  the  advantajies  of  the  great  circle 
route  could  be  apijliel  in  passing  from  one  i)olar  hemisphere  to  the 
otlier.  liiasmiK-li  as  the  radius  with  whicli  each  parallel  is  desiMibed  iu 
the  gnomonic  projection  with  the  point  of  contact  at  the  pole  is  e([ual 
to  the  tangent  of  the  co-latitude  of  that  parallel,  and  as  the  co  latitude 
of  the  e(|uator  is  90°,  the  radius  with  whi(;h  the  e<{uator  would  be 
described  is  tan  O0°=cc  .  Tlie  delineation  of  tlie  ecpiator  and  of  regions 
near  the  equator  is  therefore  impossible  when  the  point  of  contact  is 
the  pole. 

hekrle's  method. 

Within  the  present  century  many  attemi)ts  have  been  made  so  to 
devise  tiie  gnomonic  i)rojection  as  to  treat  conveniently  the  problem 
of  great  circle  sailing.  V^arious  successful  applications  were  made  from 
time  to  time,  as  seen  in  the  charts  of  Mr.  (xodfray  and  Lieutenant  llil- 
leret.  No  one  suc('eeded,  however,  in  making  the  projection  itself  an 
embodiment  of  the  means  for  measuring  both  courses  and  distances 
until  Mr.  (Justave  Herrle  designed  his  great  circle  compass  and  sug- 
gested the  application  of  the  perpendi(;ular  distance  from  the  point  of 
contact  to  the  great  circle  track  as  an  argument  in  measuring  distances 
on  the  chart  itself. 

The  princiides  of  the  Mercator  projection  are  such  that  great  circles 
on  the  spliere  will  not  generally  api)ear  as  straiglit  lines  on  the  chart, 
but  any  straight  line  on  the  chart,  excepting  a  meridian  of  longitude 
or  a  parallel  of  latitude,  represents  a  rhumb-liue  and  indicates  a  partic- 
ular mutual  bearing  of  two  places  so  connected.  On  the  sphere  such  a 
line  is  known  as  a  loxodromic  curve,  and  it  jiossesses  the  property  of 
cutting  the  meridians  at  equal  angles,  so  that,  iu  pursuing  a  straight 
line  on  the  Mercator  chart,  the  course  is  constant,  though  the  route  is 
really  circuitous.  The  simplicity  of  the  methods  necessary  for  navigat- 
ing this  circuitous  track  and  the  long  duration  of  its  usage  have  so 
intrenched  it  in  tlie  estimation  of  mariners  that  no  method  of  handling 
charts  not  analogous  to  these  has  found  favor  with  them. 

Another  essential  consideration  in  the  construction  of  charts  for  great 
circle  sailing  is  a  method  that  afllbrds  facilities  for  measuring  the  course 
and  distance  from  the  actual  i)Osition  of  the  vessel  indei)endent]y  of  any 
great  circle  track  that  may  have  been  previously  laid  down.  Just  as  the 
rhumb  course  and  distance  are  measured  on  the  Mercator  chart  from 
the  actual  ]>ositioii  in  which  the  vessel  is  found  to  be.  Both  of  these 
principles  are  recognized  by  Mr.  Herrle  in  the  construction  of  his  great 
circle  sailing  charts.  With  a  notable  degree  of  i)ublic  spirit  he  gave 
to  the  Hydrographic  Ottice  his  plans,  as  they  existed  in  1881,  and  he 
has  contributed  largely  to  the  subse<juent  maturing  of  them  and  their 
publication  in  the  form  of  the  present  excellent  great  circle  sailing 
charts  issued  by  the  Hydrographic  Office. 


28 

MEASUREMENT  OF   GREAT  CIRCLE  DISTANCES   ON    GNOMONIC   CHARTS. 
P 


Let  tte  plane  AB  represent  the  plane  of  projection  tangent  to  the 
sphere  whose  center  is  O  at  the  point  c  in  lat.  cp,  and  long.  A'.  Let 
a'h'  represent  any  great  circle  on  the  sphere,  ab  its  projection  on  tue 
tangent  plane. 

06'=  r  is  a  radius  of  the  sphere  and  is  perpendicular  to  the  plane  AB 

The  angular  distance  of  a  from  h  is  obviously  equal  to  the  angle  at 
O,  included  between  the  lines  Oa  and  Oh. 

Draw  0))i  perpendicular  to  ab.  This  will  divide  the  triangle  Oab 
into  two  right-angled  triangles,  Oam  and  Obm,  in  which 

am=Om  tan  a,  andbm=Om  tan /5, or  am-\-hm=ab=Om  (tan  a  +  tan/J). 

Draw  c7n,  connecting  the  point  of  tangency  c  with  the  foot  of  the  per- 
pendicular Om,  and  let ;/  represent  the  angle  subtended  by  cm  at  the 
center  of  the  sphere.  Since  Oc  is  i)erpendicular  to  the  plane  of  projec- 
tion it  will  be  perpendicular  to  cm,  which  is  a  line  in  that  plane,  therefore 

Om=r  sec  ;/  (2) 

Substituting  the  value  of  Om  in  (])  it  becomes 

am-\-hn=ab=r  sec  y  (tan  ^a'+tan  p)  [3) 

From  equation  (3)  it  appears  that  on  the  gnomonic  projection  the 
length  of  every  great  circle  arc,  and  consequently  every  arc  of  a  merid- 
ian, is  equal  to  the  algebraic  sum  of  t^vo  tangents  corresponding  to  a 


29 


certain  rndius,  Om  =  r  sec  ;/,  in  which  y  is  the  aiijjle,  at  the  center  of 
the  sphere,  whicli  is  subtended  by  the  perpendicuhir  from  the  point  of 
contact  upon  the  projected  great  circle  arc.  Let  a  great  circle,  ab,  be 
drawn  in  any  position  on  the  chart  (Qgiue),  mark  the  point  7n,  and  with 


c  as  a  center  and  cm  as  a  radius  describe  a  circumference.  The  two 
meridians  in  longitude  Ai,  and  A^  will  be  tangent  to  this  circumference 
at  w<i,  and  m,,  respectively.  These  three  great  circles,  ab,  PAi,  and  PA2, 
have  this  principle  in  common,  that  acertain  length  laid  off  on  each  from 
m,  nil,  and  m^  will  correspond  to  the  same  angular  distance  on  all  three. 
This  principle  will  also  be  true  with  reference  to  any  other  great  circle 
track  drawn  tangent  to  the  circumference  m,  nii,  mi. 

If  a  line  of  tangents,  AB,  be  drawn  corresponding  to  the  radius  Oc= 
r,  the  assumed  radius  of  the  sphere  upon  which  the  projection  is  con- 
structed, and  the  distance  cm,  the  perpendicular  distance  from  the 

A        A 


point  of  tangency  to  the  great  circle  arc,  be  laid  off  upon  it,  we  shall 
have  the  true  length  of  Om=r  sec  y  (see  equation  3).     If  now  a  second 


30 

line  of  tangents,  A'  B',  be  drawn  corresponding  to  the  radius  Om  =  r  sec. 
y,  and  the  distances  am  and  hm,  which  are  the  distances  from  the  ex- 
tremities of  the  great  circle  arc  to  the  foot  of  the  i^erpendicalar  from  the 
point  of  tangency  to  that  arc,  be  laid  off  upon  it,  we  shall  have  the 
values  of  angles  a  and  fi,  whose  algebraic  sum  constitutes  the  great 
circle  distance.  Upon  this  principle  the  scales  for  the  measurement  of 
distances  on  the  great  circle  sailing  charts  of  the  U.  S.  Hydrographic 
Office  are  constructed. 


MEASUREMENT   OF   GREAT  CIRCLE   COURSES   ON   GNOMONIC   CHARTS. 

On  the  gnomouic  chart  the  problem  of  finding  at  any  point  of  a  ship's 
track  that  course  which  would  lead  most  directly  to  the  point  of  des- 
tination resolves  itself  into  finding  the  relation  between  the  angle  a 
formed  by  the  straight  lines  ah  and  al?  on  the  plane  of  projection  AB 
and  the  spherical  angle  aj  formed  on  the  sphere  by  the  great  circle 
arcs  rti6i  and  «iPi. 

In  the  figure,  let  O  be  the  center  of  the  sphere  5  AB  the  plane  of  pro- 
jection, tangent  to  the  sphere  at  the  point  c ;  Pi  the  pole  of  the  sphere, 
projected  at  F  on  the  plane ;  Uibi  any  great  circle  on  the  sphere,  pro- 
jected at  ab  on  the  plane ;  ip  the  angle  at  the  center  of  the  sphere,  sub- 
tended by  the.  straight  line  connecting  the  point  of  contact  with  the  ex- 
tremity of  the  projected  great  circle. 


Suppose  the  center  O  and  the  plane  AB  to  remain  stationary  and  the 
sphere  to  be  revolved,  so  that  the  pole  Pi  will  be  projected  at  a  and  the 


(;RI;AT  CIRCLK    sailing    (mart  of   TUE    INDIAN    OCKAX 


o 


•    • 


31 

middle  Tiieridiaii  P|r  ;il<)ii<;'  the  liiu'  passiii<;-  tliroii^li  ac  In  tlic  new 
])0.siti()ii  of  tlie  s])hen'  the  latitiuU^  of  tlie  point  of  contact  will  be  !)()^ 
uiiuus  the  angular  distance  of  r  from  a.  (h\  as  cn  —  r  tan  //•.  <•  will  he 
in  hit.  00^-/. 

Ivejiardinji;  now  a  as  the  pole,  and  nieasurinf^'  distances  ot  IK)  each, 
from  (I  to  L",  L',  and  L",  the  points  L",  L',  and  L"  will  be  the  intersec- 
lions  of  the  meridians  (u\  oh,  ami  aV  with  the  projection  of  the  Cipia- 
tor  in  the  new  position  of  the  s[)here;  dl/'  will  l)e  the  middle  meridian 
and  <t\j'  and  <i\j"  will  rejnesent  the  meridians  in  loni^itmles  L'  and  h". 
Their  difference  of  lonuitude  L"  — L'  will  be  the  value  of  the  true  angle  <r\. 
Thus  it  appears  that,  if  a  net  ol"  the  meridian  lines  of  a  i)rojection  having 
its  point  of  contact  in  latitude  (!K)0— //•)  were  prepared,  the  value  of  <r, 
could  be  readily  found  by  layinji'  the  middle  meridian  of  the  net  over 
the  line  ac  with  the  i)ole  over  d,  and  readin.u'  olf  the  ditferenc^e  of  longi- 
tude between  the  lines  aV  and  <ih.  But,  as  the  distance  >ic  nuiy  vary 
from  0"  to  90",  a  great  number  of  such  nets  would  be  lecpiired  to  meet 
every  case.  A  method  of  combining  them  was  therefore  devised,  which 
has  resulted  in  the  production  of  the  great  circle  (compasses  or  course 
indicators  which  appear  on  the  great  circle  sailing  charts  of  the  Ilydro- 
grai)hic  Office.  This  consists  of  a  series  of  eijuidistant  concentric  cir- 
cumferences, which,  in  the  gnomonic  chart  of  the  Indian  Ocean,  has 
for  its  center  the  i)oint  of  contact.  I'^ach  circumference  is  divided  into 
3(>()  degrees,  in  such  a  manner  that  if  each  division  of  any  circumfer- 
ence, as  for  example  the  one  marked  latitude  36°,  were  connected  by 
radii  with  the  common  center,  the  comidete  radiation  of  the  meridians 
from  degree  to  degree  would  be  shown  for  a  gnomonic  projection  having 
its  i)oint  of  contact  in  latitude  (90'^— 30°)  or  54°.  The  circumferences 
are  marked  in  the  direction  of  tho  radius  from  3  to  3  degrees  for  values 
of  i/',  and  by  reading  from  the  chart  the  latitude  of  a  the  circumference 
which  is  to  be  used  in  the  measurement  of  the  angle  <r  is  known  at 
once,  instead  of  using  this  course  indicator  in  the  manner  described 
for  the  use  of  the  transparent  net  of  meridians,  greater  convenience  is 
attained  by  engraving  it  in  a  convenient  position  on  the  chart  plate 
and  transferring  the  point  from  which  the  course  is  to  be  measured  to 
the  center  of  it. 

In  1892  a  much  simplifu'd  great-circle  (course  diagram  was  constructed, 
and  80  applied  to  the  series  of  (ireat  Circle  Sailing  charts  of  the  Hydro- 
graphic  Office  as  to  provide  for  the  measurement  of  great  circle  courses 
from  a  compass-rose  in  nearly  the  same  manner  as  on  the  Mcrcator 
chart  and  with  nearly  ecpial  convenience. 

The  problem  of  finding  the  initial  course  on  a  great  circle  track  pass- 
ing between  two  known  geographical  positions  may  be  stated  thus: 
Given  the  two  sides  and  their  included  angle  in  a  spherical  tiiangle  to 
find  the  angle  o))i)osite  to  a  known  side.  In  the  following  figure  let  the 
arc  '/j  S  re[)resent  a  great  circle  track  joining  the  i)oint  of  de[)arture  Z 
with  the  point  of  destination  S,  and  let  I*  represent  the  position  of  the 


32 

elevated  pole  of  the  earth,  then  PZ  and  PS  will  represent  arcs  of  the 
meridians  of  longitude  passing-  through  Z  and  S  respectively,  the  angle 
ZPS  will  represent  A,  the  difference  oi"  longitude  between  Z  and  S,  and, 
if  (^1  is  the  latitude  of  Z  and  (pi  the  latitude  of  S,  the  length  of  the 


sides  PZ  and  PS  of  the  spherical  triangle  PZS  will  be  90°— ^i  and 
90^— f/: 2  respectively. 

From  the  fundamental  formuhe  of  spherical  trigonometry  we  have 

siu  (90°- ^i)  cot  (90^— ^.>)=cot  C  sin  A  +  cos  (90O  — 9^,)  cos  A     (1) 
or  cos  qjx  tan  ^2= cot  C  sin  A  +  sin  q)\  cos  A  (2) 

whence  cot  q^cos  9^  tan  ^2-sin  y,  cos  A 

sin  A 
Dividing  both  terms  of  the  fraction  of  the  second  member  by  cos  <^,, 

cot  C— ^^°  (^2— tan  ^1  cos  A  ^. 

sec  (p  I  sin  A 

,  t^    tan  (B2 — tancoiCosA  ,r\ 

or  cot  0=       /Tf . — T~-^ —  (^) 

sec  q)i  sm  A  —  0 

Putting  2/= tan  cpx  cos  A  ^ 

y'=Uucp2  I  ,Q. 

ir=sec<^i  sin  A    j  ^  '' 

x'=o  J 

equation  (5)  becomes 

cotO=-^"~^'  (7) 

x — o 

Therefore  G  is  the  angle  made  with  the  axis  of  Y  by  a  straight  line 
joining  the  points  {,v,  y)  and  (o,  y'). 
From  equation  (G) 

.     - = sec r«  1  and     -  ,=  tan  <w  1 
siuA  cos  A 

or    /„■■ ?L.^=sec'^r/;,  — tan'r/'^2=l  (8) 

sm^A    cos^A 

which  is  the  equation  to  a  hyperbola  whose  semiminor  and  semimajor 
axes  are  sin  A  and  cos  A,  respectively. 


CKKAT  CIHCI.E  SAII.IXc;  (  IIAKl    111     I  1 1  K  NORTH  PA(II''IC  OCEAN 


is 


SI   I'PI  I  MI  ^  lAIlY      MITimn      I  Oli      I  IM)1N(.      (OIKSCS 


iiiiiiiiiiiiiliiiBli 


ii\-iH--i--"--,-i--r-r-i'i'r-"-"-r-i--i~'Ti'  ■■•■- 


"■'•'•"'  ■  ' -  i"i jiii-jriiyB^fyyy syi!! . !  as.si=tsS'ss^&ss^^?is»ii 


....BSSiliiiiBS ..  I   aagB^5Ji59g-g5agag4g'.a 


li 


33 

Therefore  the  point  (.r,  y)  will  always  be  found  upon  the  arc  of  a 
hyperbola  upon  whose  major  axis  the  corresponding  point  (o,  y')  is 
found  with  reference  to  a  scale  of  tangents  representing  ?/'= tan  (fj-^. 


X 


In  the  construction  of  the  course  diagram  of  tnc  accompanying 
great  circle  chart  of  the  North  Pacitic  Ocean,  A  was  assumed  to  be 
ecjual  to  20^.  This  explains  why,  in  the  directions  for  measuring 
courses,  a  point  is  always  selected  on  the  great  circle  track  at  an  inter- 
val of  20'^  of  longitude  from  the  point  of  departure. 


19802- 


-3 


SECTION  III. 


MISCELLANEOUS  METHODS. 
airy's  method. 

Sir  George  Airy,  late  astronomer  royal  of  Great  Britain,  has  devised 
the  following  graphical  method  of  describing  upon  a  Mercator  projection 
that  circnlar  arc  which  shall  approximate  most  nearly  to  the  projection 
of  a  great  circle  of  the  globe. 

Draw  the  rhumb-line  AB  between  the  place  of  departure  and  the 
place  of  destination,  and  the  perpendicular  j)P_at  the  middle  point  of 
AB.  With  the  middle  latitude  between  the  two  places  enter  the  fol- 
lowing table  and  take  out  the  corresponding  parallel.  The  center  of 
the  required  circular  arc  will  be  at  the  intersection  of  this  parallel  with 
the  perpendicular. 


Middle 

Corresponding 

Middle             Corresponding 

latitude 

parallel. 

latitude.                  parallel. 

0 

0           ' 

0                                                      O           ! 

20-1 
22 
24 

81     13 
78     16 
74    59 

Ill     Opposite 
56)  .      °'*°"^-        1 

11     33        1 
6     24 
1     13 

26 

71     26 

58 

4    00 

28 

67     1!8 

60 

9     15 

30 

63    37 

62 

14    32 

32 

r>9    25 

64 

19    50 

34 

Opposite 

.-'5    05 

66 

25    09 

36 
38 

name. 

50    36 
46    00 

68 

70 

'  Same  iiaiue.  > 

30    30 
35    52 

40 

41     18 

72 

41     14 

42 

36    31 

74 

46    37 

44 

31     38 

76 

52     01 

46 

26    42 

78 

57    25 

48 

21     42 

80 

62    51 

50 

16    39 

1 

34 


35 

The  above  table  is  taken  from  ''  Notes  on  Navigation  aud  the  Deter- 
mination of  Meridian  Distances,"  by  Commander  P.  F.  Ilarrington,  U. 
S.  Navy. 

The  center  of  the  arc  to  be  described  frequently  falls  beyond  the 
limits  of  the  chart  which  may  be  employed,  and  the  track  may  occupy 
more  than  one  chart.  In  these  cases  diflficnlties  arise  in  drawing  the  re- 
quired arc. 

It  is  of  value  to  project  a  great  circle  track  to  which  it  is  intended  to 
adhere  throughout  a  passage,  but  if  a  vessel  be  diverted  from  the  pro- 
jected great  circle  a  new  track  must  be  laid  down  with  the  attendant 
inconveniences.  Great  circle  distances  can  not  be  measured  closely  on 
the  approximate  arc. 

fisher's  method  for  circular  arc  sailing. 

Besides  the  foregoing  there  is  another  method  for  describing  circular 
arcs  as  substitutes  for  actual  great  circle  tracks  on  sailing  charts. 
This  method  was  proposed  by  the  Rev.  George  Fisher,  M.  A.,  F.  R.  S., 
and  described  in  the  article  on  circular  arc  sailing  in  Riddle's  Naviga- 
tion, published  in  18G4.  Unfortunately  it  fiuds  its  most  extensive  and 
easy  application  in  those  cases  in  which  the  vertex  of  the  gr«at  circle 
is  in  so  high  a  latitude  as  to  make  the  navigation  of  it  dangerous.  In 
the  tropics  the  method  is  impracticable,  because  great  circular  arcs  are 
there  represented  on  sailing  charts  nearly  as  straight  lines ;  and  in 
the  region  between  40°  and  00°  of  latitude  the  curvature  of  the  great 
circular  arcs  on  sailing  charts  is  so  small  that  it  is  ditficult  to  describe 
substitutes  for  them  on  account  of  the  length  of  the  radius  required. 


The  curve  PRQAVB  shows  the  direction  of  the  great  circle  which 
passes  through  the  points  A  and  B,  which  represent  the  Capo  of  Good 
Hope  and  the  south  part  of  Van  Diemen's  Land.    Although  in  sailing 


36 

from  one  place  to  another  the  navigator  is  only  coucernetl  with  that 
portion  of  the  great  circle  upon  which  he  means  to  travel,  the  curve 
is  nevertheless  continued  through  both  hemispheres  in  the  figure,  in 
order  that  its  general  form  may  be  the  better  comprehended.  The 
northern  and  southern  portions  of  this  curve  are  equal  and  similar  to 
each  other;  and  the  curve  cuts  the  equator  at  two  points,  P  and  Q, 
at  a  distance  of  180°  of  longitude  from  each  other,  an<l  at  an  angle, 
which,  at  the  point  of  intersection  or  contrary  flexure,  is  equal  to  the 
latitude  of  the  vertex  or  the  inclination  of  the  planes  of  the  equator 
and  the  great  circle.  In  like  manner  the  curve  PV"  QV  represents 
another  great  circle  projected  upon  the  chart,  the  highest  point  of  lati- 
tude V"  or  V  being  80°. 

The  near  approximation  to  a  circular  form  of  the  portion  of  the 
curve  which  is  nearest  to  the  vertex  aftbrds  an  easy  and  simple  mode 
of  delineating  upon  a  Mercator  chart  a  ship's  route  be.tweeu  two  places 
thus  situated. 

It  consists  in  finding  the  position  of  the  vertex  of  the  great  circle 
which  passes  through  the  points  of  departure  and  destination,  and  de- 
scribing as  the  track  to  be  pursued  a  circular  arc  passing  through  these 
three  places.  Upon  this  arc  the  geographical  positions  of  any  number 
of  points  can  be  determined  to  a  degree  of  accuracy  which  will  depend 
upon  the  scale  of  the  chart,  which  known  points  can  be  sailed  toward 
in  succession  by  either  using  Middle  Latitude  or  Mercator's  Sailing. 

When  the  great  circle  track  passes  to  a  higher  latitude  than  the  nav- 
igator wishes  to  attain,  Mr.  Fisher  proposes  a  circular  track,  which  shall 
pass  through  the  three  points  representing  the  places  of  departure,  des- 
tination, and  the  highest  point  which  it  is  desirable  to  attain. 

In  the  practical  application  of  this  method,  instead  of  drawing  the 
whole  of  a  circular  arc  between  the  places  of  departure  and  destination, 
it  is  necessary  only  to  assume  one  point  upon  the  arc  at  a  convenient 
distance  from  the  place  of  departure.  Then  determine  its  position  upon 
the  chart,  and  shape  a  course  toward  it.  Should  the  ship's  position  by 
subsequent  observations  be  found  to  be  off  the  circular  arc  previously 
determined,  instead  of  trying  to  return  to  it  another  circular  arc  should 
be  described  upon  the  chart  extending  from  the  ship's  position  to  the 
place  sailed  for,  and  another  point  taken  upon,  as  before. 

chauvenet's  great  circle  protractor. 

By  the  use  of  this  protractor  the  latitudes  and  longitudes  of  all  points 
on  the  globe  through  which  the  great  circle  route  passes,  and  also  the 
course  to  be  steered  and  the  distance  to  be  sailed  are  found  by  inspec- 
tion. It  consists  of  two  stereographic  projections  of  the  spherical  rec- 
tangular co-ordinates  on  the  same  scale,  one  fixed  and  the  other  trans- 
parent, and  designed  to  revolve  concentrically  with  the  first.  The  fixed 
])rqiection  represents  the  meridians  and  parallels  of  latitude  of  the 
globe  to  every  degree.    No  laud  is  shown,  so  that  any  meridian  may  be 


37 

assumed  as  the  meridian  of  the  place  of  departure.     For  simplicity  the 
primitive  or  bouudiug  meridian  is  always  so  taken.     In  tUe  tigure  the 


full  lines  and  the  lines  of  dashes  represent  the  lines  on  the  fixed  and 
transparent  projections,  respectively.  The  lines  corresponding  to  the 
meridians  on  the  revolving  projection  evidently  represent  a  complete 
system  of  great  circles  with  reference  to  CD  as  the  equator,  while  the 
lines  corresponding  to  the  parallels  are  lines  which  divide  the  great  cir- 
cles of  the  system  into  sections  of  equal  length,  and  hence  may  be  called 
distance  lines.  The  line  GH  of  the  fixed  projection  is  graduated  from 
zero  at  G  and  zero  at  II  to  90^  at  O,  and  forms  the  scale  of  longitndcs. 
The  corresponding  line  EF  of  the  revolving  projection  is  similarly  grad- 
uated, and  forms  the  scale  of  courses.  Since  the  inclination  at  G  of 
each  meridian  to  the  primitive  must  be  equal  to  its  longitude  from  the 
primitive,  the  lino  AG  of  the  fixed  projection  is  graduated  from  zero 
at  G  to  90O  at  A  and  at  B.  The  corresponding  line  CO  of  the  revolving 
projection  has  a  double  graduation  proceeding  from  90^  at  G  to  zero  and 
180O  at  C  and  at  D. 

To  find  the  great  circle  route  between  any  two  places,  C  ami  K,  on  the 
fixed  projection,  i)lot  the  position  of  G,  the  point  of  dei)arturo,  on  the 
primitive  meridian  by  means  of  its  latitude,  and  the  position  of  Kin  its 
proper  latitude  on  that  meridian  of  the  fixed  projection,  whose  angular 
distance  from  the  primitive  is  eipial  to  the  difference  of  longitude  be- 
tween G  and  K.  Revolve  the  transparent  projection  until  the  western 
common  intersection  of  the  system  of  great  circles  coincides  with  the 
point  of  departure  if  that  point  is  on  the  western  side  of  the  fixed  chart, 
or  untd  the  eastern  common  Intersection  of  the  system  of  great  circles 
coincides  with  the  point  of  departure  if  it  is  on  the  eastern  side  of  the 
fixed  chart.    The  great  circle  which  passes  through  K  is  the  one  re- 


38 

quired.  The  reading  of  its  intersection  with  EF  (see  figure)  is  the  first 
course.  The  distance  in  nautical  miles  between  any  two  points  on  the 
great  circle  is  found  by  observing  the  distance  lines  which  pass  throngh 
these  points  and  multiplying  the  difference  of  their  readings  on  the 
scale  CD  (see  figure)  by  60'. 

In  some  positions  of  the  protractor  such  a  maze  of  lines  is  formed  as 
to  give  rise  to  great  indistinctness  and  consequent  difficulty  in  making 
the  required  reading.  This,  added  to  the  liability  of  the  transparent 
material  to  become  warped  and  brittle,  and  the  difficulty  of  keeping 
the  two  projections  concentric,  has  caused  its  disuse. 

sigsbee's  great  circle  protractor. 

It  has  been  pointed  out  that  the  objectionable  feature  of  Chauvenet's 
Great  Circle  Protractor  is  the  transparent  revolving  projection.  In  the 
present  method,  which  was  pro.luced  by  Commander  C.  13.  Sigsbee, 
U.  S.  I^avy,  in  1885,  this  defect  has  been  overcome  by  the  adaptation 
of  a  single  projection  or  system  of  circles  to  the  measurement  of  more 
than  one  system  of  coordinates  in  the  solution  of  the  same  problem. 
This  method  is  capable  of  the  approximate  solution  of  spherical  prob- 
lems in  general,  but  it  will  be  treated  of  here,  so  far  as  it  relates  to 
the  solution  of  problems  in  great  circle  sailing  only.  The  following 
description  is  transcribed  from  an  article  entitled  "Graphical  Method 
for  Navigators,''  by  Commander  Sigsbee,  U.  S.  Navy,  printed  in  the 
Proceedings  of  the  United  States  Naval  Institute,  Vol.  XI,  No.  2, 1885; 


M" 


In  the  figure,  let  us  suppose  that  the  full  lines,  consisting  of  a  system 
of  great  circles  and  parallels  on  the  stereographic  projection,  with  M' 
and  M"  as  the  poles  of  the  latter,  have  served  to  project  the  points  M 
and  N,  the  co-ordinates  of  which  are  given. 


39 

Let  us  suppose  further  that  M  is  one  of  the  poles  of  a  similar  set  of 
great  circles,  and  that  to  this  system  of  dotted  lines,  which  serve  for  the 
measurement  of  auother  system  of  spherical  co-ordinates,  the  point  N 
is  to  be  referred  for  the  solution  of  the  problem.  To  make  the  explana- 
tion more  practical,  let  us  assume  first,  that  the  full  lines  consist  of 
vertical  circles  or  circles  of  azimuth  and  i)arallels  of  altitude,  the 
primitive  or  bounding  circle  being  the  meridian  of  the  observer,  M' 
his  zenith,  and  that  having  the  azimuth  and  altitude  of  a  heavenly  body 
given,  we  have  projected  its  place,  N,  upon  the  figure;  second,  that  i\r 
is  the  elevated  celestial  pole,  the  dotted  lines  hour  circles  and  parallels 
of  declination,  and  that  we  require  the  hour  angle  and  declination  of 
the  body  at  N. 

If  the  graduations  of  the  primitive  aud  of  the  dotted  diameter  or 
equinoctial  CD  were  properly  numbered  we  would  simply  have  to  note 
the  parallel  of  declination  and  the  hour  circle  passing  through  the  point 
N,  follow  one  to  the  primitive  and  the  other  to  the  equinoctial,  and 
take  readings,  in  order  to  find  the  declination  and  the  hour  angle  of  the 
body,  that  is  to  say,  the  co-ordinates  of  the  point  N,  according  to  the 
system  of  co-ordinates  measured  by  the  dotted  lines.  So  long  as  the 
relative  positions  of  the  two  points  M  and  M'  remained  the  same  such 
a  figure  would  serve  for  the  solution  of  similar  problems  involving  any 
other  position  of  an  interior  point,  IsT ;  but,  since  the  relative  positions 
of  M  aud  M'  are  constantly  changing  in  practice,  no  two  sets  of  lines 
similar  to  those  of  the  figure,  and  printed  upon  a  single  sheet,  can  be 
of  universal  application  in  the  manner  described.  The  object  of  the 
present  method  is  to  overcome  this  difficulty  by  adapting  the  system  of 
full  lines  to  serve  the  purpose  of  both  systems  for  all  positions  of  M 
aud  N. 

The  position  of  N,  with  respect  to  the  dotted  lines,  is  defined  by  its 
position  relative  to  the  points  M  and  O.  Since  the  two  systems  of  lines 
are  similar,  if  we  transfer  IST  to  N'  so  that  N'  shall  have  the  same  posi- 
tion with  respect  to  M'  and  O  that  ]S^  has  with  respect  to  M  and  O,  then 
N'  will  have  the  same  relation  to  the  full  lines  that  N  has  to  the  dotted 
lines,  and  we  may  therefore  let  N'  represent  JST,  and  the  system  of  full 
lines,  iu  connection  therewith,  represent  the  system  of  dotted  lines. 
Briefly,  then,  the  method  is  to  assume,  first,  that  the  full  lines  repre- 
sent a  system  of  spherical  co-ordinates  to  correspond  with  the  given 
data,  and  then  to  project  M  and  IST ;  next,  to  transfer  N  to  N',  and  then 
to  assume  that  the  same  lines  represent  another  system  of  spherical 
co-ordinates,  to  which  it  is  necessary  to  refer  to  N'  for  a  solution. 

The  following  elementary,  graphical  process  forms  the  basis  of  solu- 
tions. Having  projected  on  the  diagram  two  points,  as  M  and  N,  given 
iu  position,  one  upon  the  primitive  or  bouiuling  circle,  and  the  other 
within,  conceive  a  sector,  MOC,  whose  radii,  OM  and  00,  shall  include 
these  points.  Oonceive  the  imaginary  sector  to  be  revolved  about  O 
until  M  coincides  with  some  other  given  point  upon  the  primitive,  as 


40 

M' ;  then  find  N',  the  revolved  position  of  N.     The  radii  need  never  he 
draicn. 

There  are  various  ways  of  finding  N',  but  the  following  are  suggested. 
The  first  is  always  available,  and  involves  marking  points  only  upon 
the  diagram;  the  second  requires  a  i)iece  of  tracing-paper,  but  makes 
no  marks  upon  the  diagram.  Since  one  case  embraces  all,  let  it  be  re- 
quired to  revolve  the  sector  MOC  about  O  until  M  coincides  with  M', 
and  find  N'.  Since  M  will  traverse  the  arc  MM',  the  point  0  will 
traverse  an  equal  arc  CA. 

First  method. — Align  a  straight  edge  on  O  and  N  to  find  the  point 
C.  Make  the  arc  CA  equal  to  the  arc  MM',  either  by  the  divisions  of 
the  scale  on  the  primitive  or  by  transferring  the  chord  MM'  to  CA  with 
a  slip  of  paper.  Align  a  slip  of  paper  on  O  and  C  and  mark  upon  it 
the  i)oints  O,  N,  and  C.  Then  align  the  slip  on  O  and  A  so  that  its 
marks  O  and  C  shall  coincide  with  O  and  A  of  the  diagram,  resi)ect- 
ively.     The  point  N  of  the  slip  will  be  2^',  the  revolved  position  of  N. 

Second  method. — Lay  a  piece  of  tracing-paper  upon  the  diagram  and 
trace  the  points  M,  O,  and  ]!f.  Eevolve  the  tracing  about  O  until  M 
coincides  with  M' ;  the  traced  point  N  will  fall  at  N'. 

The  adaptation  of  the  method  to  great  circle  sailing. — Let  M  be  the  place 
of  departure,  and  N  the  place  of  destination.  To  find  the  great  circle 
course,  first  assume  the  diagram  to  be  the  projection  of  the  terrestrial 
sphere,  composed  of  parallels  of  latitude  and  meridians  of  longitude. 
M'  is  the  north  pole,  M"  the  south  pole,  and  AB  the  equator.  The 
primitive  is  always  the  meridian  of  departure.  Project  M  upon  the 
primitive  in  its  proper  latitude,  north  or  south,  as  the  case  may  be — on 
the  right  side  if  it  is  the  eastern  place,  on  the  left  side  if  it  is  the  west- 
ern place.  Project  N  in  its  ])roper  latitude,  and  upon  a  meridian  whose 
difference  of  longitude  from  M  is  that  of  the  two  places.  Conceive  a 
sector,  MOC,  formed  by  radii,  to  include  M  and  N.  Note,  by  a  glance 
simply,  if  N  would  fall  above  or  below  AB  if  the  sector  were  revolved 
so  as  to  make  M  coincide  with  that  extremity  of  AB  which  is  adjacent 
to  M.  If  above,  reckon  the  course  from  north;  if  below,  from  south. 
Revolve  the  sector  about  O  until  M  coincides  with  M'  or  M",  the  nearer 
extremity  of  M'  M",  and  find  N',  the  revolved  jjosition  of  N.  Now,  as- 
sume M'  or  M",  whichever  is  the  revolved  position  of  M,  to  be  the  place 
of  departure  and  N'  the  place  of  destination.  The  former  meridians 
then  become  great  circles  through  the  place  of  departure,  and  the 
parallels  are  parallels  of  great  circle  distance  from  the  same  place 
The  scale  AB  gives  the  angle  which  each  great  circle  makes  with  the 
primitive,  the  meridian  of  departure,  and  hence  the  course. 

The  great  circle  passing  through  M',  N',  and  M"  is  the  required  great 
circle.  Read  the  course  at  its  intersection  with  AB,  reckoning  from  the 
nearer  extremity  of  AB.  Having  the  course,  reckon  it  from  north  or 
south,  as  previously  found,  and  towards  the  east  or  west,  as  the  place  of 
destination  is  to  the  eastward  or  westward. 


• 


r-t==55'"  r     T^ 


Im 


41 

To  find  the  great  circle  distance. — Note  the  reading  upon  the  primitive 
at  the  parallel  of  distance  passing  tlirongh  N',  reckon  the  distance  from 
M'  or  M",  the  place  of  dejjarture,  by  taking  the  complement  of  the 
reading.  Multiply  the  degrees  by  60  and  add  the  minutes;  the  result 
will  be  the  distance  in  nautical  miles. 

To  find  the  vertex  and  other  point. s  upon  the  great  circle. — The<iuickest 
method  is  by  means  of  tracing  i)aper.  Trace  the  required  great  circle 
through  N',  and  revolve  the  tracing  about  ()  until  M'  or  M",  whichever 
is  the  revolved  positi(jn  of  M,  coincides  with  M.  The  traced  great  circle 
will  then  pass  through  M  and  N.  If  the  vertex  is  of  any  use  it  falls 
upon  the  diagram,  and  it  is  found  upon  a  meridian  at  90^  difference  of 
longitude  from  the  point  where  the  traced  and  revolved  great  circle 
intersects  AB,  the  ecpiator. 

Take  points  upon  the  revolved  great  circle  at  5^  or  10°  intervals  of 
longitude  from  M  towards  N,  or,  if  desired,  on  both  sides  of  the  vertex 
when  it  falls  between  M  and  N,  and  And  the  latitude  and  longitude  of 
each,  measuring  latitudes  upon  the  primitive  and  difi'erences  of  longi- 
tude from  M  upon  the  scale  of  AB.  Transfer  the  points  to  the  sailing 
chart  and  adjust  or  "  fair"  a  curve  to  them. 

The  more  exact  method  is  to  take  the  intervals  from  M  toward  K, 
for  the  points  will  then  fall  upon  printed  meridians  of  the  diagram. 
The  advantage  of  measuring  from  the  vertex  is  that  points  ecjually 
distant  in  longitude  on  either  side  have  the  same  latitude. 

HARRIS'S*   METHOD    OF    USING    A    STEREOGRAPHIC    PROJECTION    IN 
THE   SOLUTION    OF   SPHERICAL   TRIANGLES. 

1.  One  of  the  diificulties  in  getting  close  results  with  either  Chauve- 
net's  or  Sigsbee's  protractor,  and  with  most  planispheric  arrangements, 
arises  from  the  fact  that  the  paper  or  other  material  upon  which  the 
projections  are  iirinted  maj^  not  contract  or  expand  uniformly  in  all  its 
parts.  For  instance,  one  diameter  of  the  printed  hemisphere  may  be  a 
degree  or  two  longer  than  the  diameter  perpendicular  to  it.  Any  proc- 
ess which  involves  actual  rotation,  about  the  center  of  the  projection, 
of  any  point  or  line  is  objectionable  unless  the  projection  be  made  upon 
a  plate  of  metal.  If  an  instrument  of  this  kind  were  contemplated,  a 
pair  of  graduated  arms  radiating  from  the  center  and  capable  of  being 
set  at  any  angle,  but  which  could  be  removed  at  pleasure,  would  be 
found  to  be  convenient.  • 

The  protractor  here  described  aims  to  do  away  with  the  objections 
just  raised  by  adding  a  set  of  dotted  lines,  representing  meridians  and 
parallels  of  an  equatorial  projection,  to  the  ordinary  meridional  projec- 
tion of  the  hemisphere.  See  Fig.  1,  in  which  the  broken  lines  are 
intended  to  consist  of  dots  one  degree  apart.  If,  therefore,  the  paper 
becomes  somewhat  distorted,  the  relation  between  the  two  sets  of  lines 
or  coordinates  will  remain  unchanged. 


*Due  to  Dr.  Kolliu  A.  Harris  of  the  U.  S.  Coast  and  Geodetic  Survey. 


42 

2.  Right-angled  triangles. 

Always  suppose  the  right  angle  to  fall  upon  the  bounding  great  circle, 
say  at  R,  Fig.  2.  (Reference  to  Figs.  2,  3,  or  4,  implies  reference  to 
Fig.  1  also.)  Suppose  one  of  the  oblique  angles  to  be  placed  at  the  pole 
N;  when  one  of  these  two  angles  is  known,  place  there  the  known  angle. 

The  value  of  the  angle  is  denoted  by  the  numbering  of  the  great  cir- 
cles. The  length  of  the  leg  along  the  limb  of  the  projection  is  denoted 
by  the  numbering  of  the  graduations  or  divisions;  the  length  of  the 
other  leg,  by  the  numbering  of  the  dotted  small  circles  (along  AB,  Fig.  1). 


This  construction  suifices  for  all  right  triangles,  excepting  the  case 
where  the  two  given  parts  are  the  oblique  angles.  In-  this  case  sub- 
tract each  of  them,  as  well  as  the  right  angle,  from  180°.  These  three 
supplements  are  the  three  sides  of  the  polar  triangle.  It  is  a  quad- 
rantal  triangle  and  falls  under  §3;  two  angles  being  thus  found,  their 
supplements  are  the  two  legs  of  the  original  triangle. 

3.  Quadrantal  triangles. 

Let  the  line  ON,  Fig.  3,  represent  the  side  whose  length  is  a  quad- 
rant. Let  jSTH  and  HO  represent  the  other  sides.  The  numbering  of 
the  dotted  small  circles  (Fig.  1)  which  measure  HO  should  increase  from 
the  center  outward ;  at  the  circumference  the  numbering  is  90°.  When 
the  side  to  be  taken  alo^g  OR  exceeds  90°,  the  numbering  increases 
from  90°  at  the  circumference  to  180°  at  the  center,  just  as  if  this  line 
were  continued  on  the  under  surface  of  the  paper.  The  angle  at  O  is 
measured  by  the  arc  NR;  the  angle  at  IsT  is  measured  along  OW,  the 
center  O  being  numbered  0°  and  180°,  the  point  W,  90°. 

This  construction  suffices  for  all  quadrantal  triangles,  except  where 
the  angle  opposite  the  quadrant  is  one  of  the  parts  considered.  This 
case  can  be  treated  by  taking  each  known  part,  including  the  quad- 
rantal side,  from  180°,  thus  obtaining  the  polar  triangle.  This  being  a 
right  triangle  falls  under  §2. 


43 

■    4.  All  cases  of  ohlUiuv  triangles^  e.vcepthnj  ichere  ihc  three  sides  or  the 
three  unfiles  arc  the  {/iren  parts. 

Form  two  right  triangles  out  of,  or  in  cuiiiiectiou  witli,  the  given 


S 
Fig.  3. 

oblique  triangle  by  letting  fall  a  perpendicular  arc  from  one  of  the 
angles  not  immediately  sought,  upon  the  opposite  side,  or  side  produced. 

Treat  the  two  right  triangles  by  §  2. 

5.  Special  treatment  for  the  ease  where  tu-o  sides  and  the  included  angle 
are  given. 

Let  NG,  Fig.  4,  denote  one  of  the  known  sides.     The  knowu  angle 

JV 


Fi^.L 

at  I«r  shows  along  what  arc  the  second  known  side  NH  is  to  be  taken. 
The  dotted  radial  line  OR  passing  through  H  has  a  delinite  number 
!N"K  assigned  to  it.  If  we  imagine  Gil  to  be  rotated  about  O  until  G 
falls  at  N,  the  angle  ENH'  will  be  equal  to  the  original  angle  NGII. 
The  arc  Nil'  will  be  equal  to  the  original  unknown  side  GET.  Instead 
of  actually  rotating  H,  the  position  H'  is  found  by  means  of  the  fixed 


44 

numbers  belonging"  to  tlie  radial  lines,  say,  by  counting  from  H  to  H'  a 
number  of  degrees  equal  to  GN. 

6.  To  find  the  node  or  vertex  of  the  (jreat  eircle  passing  through  two 
given  points. 

In  the  right  triangle  (tWI,  the  side  GrW  and  the  angle  HGW  are 
known  by  the  preceding  solution  of  the  triangle  IS^GH  ;  this  being  a 
right  triangle,  comes  under  §  2.  WI  denotes  the  longitude  of  the  node 
from  W,  and  the  angle  HIW,  the  latitude  of  the  vertex.  The  longitude 
of  the  vertex  is,  of  course,  90'^  from  I. 

7.  To  construct  the  track. 

For  ascertaining  the  points  through  which  the  great  circle  track 
passes,  imagine  the  dotted  lines  of  Fig.  1  to  represent  meridians  and 
parallels.  The  system  of  great  circles  (full  lines)  represent  paths  inter- 
secting at  all  i)ossible  angles  with  the  equator.  The  orthogonal  system 
of  small  circles  serve  as  distance  lines.  The  longitude  of  the  node  hav- 
ing been  found  as  above,  also  the  latitude  of  the  vertex,  each  point  of 
the  track  be;'omes  known  in  latitude,  longitude  (from  the  node),  and 
distance.  The  bearings  of  the  intermediate  points  are  the  angles  made 
by  the  track  and  the  dotted  meridian  lines;  hence  they  can  be  roughly 
measured  or  estimated.  When  both  the  northern  and  southern  hemi- 
spheres are  concerned  in  the  same  problem,  we  can  imagine  the  same 
projection  or  sets  of  lines  on  the  under  side  of  the  jiaper  as  on  the 
visible  side. 

8.  Oiven  the  three  sides  of  a  triangle  to  find  tivo  of  the  angles. 

In  order  to  facilitate  the  solution  of  the  problem  it  may  be  well  to  make 
use  of  a  .smaller  stereographic  projection  than  Fig.  I  for  obtaining  the 
first  approximate  results.  In  connection  with  this  there  should  be  used 
a  transparent  sheet  upon  which  any  given  small  circle  can  be  traced 
(Sigsbee's  method),  or  upon  which  is  traced,  once  for  all,  the  entire 
projection  (Chauveuet's  method). 

Lay  off  the  side  opposite  the  angle  not  sought  along  the  bounding 
circle  of  the  protractor,  counting  distance  from  one  of  the  poles,  Mark 
the  two  distance  lines  (small  circles)  whose  numbers  denote  the  lengths 
of  the  two  remaining  sides.  Now  if  one  of  these  distance  lines  be 
traced  upon  tracing  i)aper,  and  if  the  line  of  centers  (central  meridian) 
be  rotated  until  it  comes  to  the  other  extremity  of  the  side  laid  down 
along  the  bounding  circle,  the  point  in  which  the  two  distance  lines 
intersect  will  be  determined.  The  number  of  the  meridian  passing 
through  this  point  shows  the  value  of  the  angle  opi)osite  the  side  cor- 
responding to  the  rotated  distance  line.  By  marking  this  point  on  the 
traced  distance  line,  and  rotating  it  back  to  the  initial  ])osition,  the 
meridian  (counted  from  the  other  limb  of  the  projection)  passing  through 
this  })oint  will  denote  the  other  angle,  which  lies  ui)on  the  bounding 
circle. 

The  more  accurate  values  are  obtained  by  means  of  the  large  pro- 
tractor, Fig.  1,  and  a  small  piece  of  tracing  paper.  From  the  small 
protractor,  if  a  small  one  has  been  used,  the  approximate  jiosition  of 


45 

the  i)oint  of  intersection  rotated  back,  becomes  known.  Place  a  i)iece 
oftracinj^  paper  over  the  large  i)rotractor,  and  trace  ui)ou  it  a  few  dots 
of  the  dotted  lines  and  a  small  arc  of  tlie  distance  line.  Rotate  it  by 
addition,  or  by  counting  an  amount  ecjual  to  the  luimber  of  degrees  iu 
the  side  laid  off  on  the  bounding  circle.  Then  mark  upon  it  accurately 
the  point  of  intersection  of  the  rotated  and  stationary  distance  circles. 
The  numbering  of  the  great  circle  passing  through  this  i)oiMt  will  denote 
one  of  the  reijuired  angles.  Carry  the  arc  thus  marked  back  to  its 
original  jiosition.     This  will  give  the  other  angle. 

Whenever  only  rough  results  are  retjuired,  a  rather  small  Chauve- 
net's  protractor  is  a  great  convenience  in  almost  all  problems  involving 
spherical  triangles.  For  instance,  a  heavenly  body  whose  hour  angle 
and  declination  are  given,  together  with  the  latitude  of  the  i)lace,  can 
be  immediately  referred  to  altazimuth  coordinates  by  inclining  the 
axes  of  the  projections  to  each  other  an  angle  equal  to  the  colatitude  of 
the  place.  Such  problems  are  solved  by  this  means  in  a  very  natural 
manner.  In  fact,  this  small  protractor  will  be  found  to  be  highly  serv- 
iceable in  laying  out  the  work  to  be  performed  upon  the  large  i^rojcc- 
tiou,  or  iu  roughly  checking  the  results. 

9.  Given  the  three  amjles  of  a  triangle  to  find  two  of  the  sides. 

Subtract  each  angle  from  180^,  thus  obtaining  the  lengths  of  the 
sides  of  the  polar  triangle.  Two  angles  of  this  triangle  become  known 
by  the  preceding  method.  These  angles  subtrjicted  from  180^  give  two 
of  the  required  sides  of  the  original  triangle. 

proctor's  method. 

By  a  comprehension  of  the  conservative  disposition  of  seafaring  men, 
and  by  a  study  of  their  wants,  the  late  Richard  A.  Proctor  arrived  at 
a  method  for  finding  the  shortest  sea  routes,  which,  comprising  nearly 
the  whole  of  the  navigable  world  within  small  compass,  is  unsurpassed 
for  the  generality  of  the  results  obtainable. 

Proctor  was  evidently  unaware  of  the  successful  issue  of  the  investi- 
gations into  the  methods  for  measuring  courses  and  distances  on  the 
gnomonic  projection.  Indeed,  these  results  have  only  lately  appeared 
on  the  great  circle  sailing  charts  of  the  l^^.  S.  Ilydrographic  Office, 
and  the  mathematical  theory  of  them  is  given  for  the  first  time  in  this 
work. 

Though  searching  in  his  investigations  for  some  projection  on  Avhich 
the  great  circle  will  appear  in  simple  geometrical  form,  he  discarded  the 
gnomonic  partly  on  account  of  the  irregular  distortion,  which  precludes 
the  possibility  of  directly  measuring  courses  by  means  of  the  ordinary 
compass  rose  or  a  protractor,  and  partly  because  he  aimed  to  show  the 
whole  navigable  world  on  one  projection,  while  on  a  single  gnomonic 
l)rojection  it  is  not  possible  to  show  one  entire  hemisphere. 

Like  other  investigators  into  this  subject,  he- saw  how  deep  rooted  in 
the  maritime  world  are  the  ways  of  handling  the  Mercator  chart,  and 
he  aimed  to  produce  a  method  which  would  aftbrd  as  great  convenience 


46 

for  pursuing  a  great  circle  track  as  the  Mercator  chart  affords  for  pur- 
suing a  rhumb.  We  therefore  find  him  seeking  a  projection  which  shall 
be  perfectly  correct  in  detail,  one  in  which  distances  over  small  areas 
are  so  correctly  proportioned  that  no  distortion  can  be  detected,  and  in 
which  all  hearings  atid  directions  are  correspondingly  correct.  He  was 
accordingly  led  to  propose  the  stereographic  projection,  whereby  (1)  the 
construction  for  marking  in  the  great  circle  track  between  any  two  points 
is  made  exceedingly  simple ;  (2)  the  whole  track  is  obtained  at  once; 
(3)  the  course  at  each  point  of  the  track  is  shown  as  plainly  as  the  bear- 
ing of  the  rhumb-line  on  a  Mercator  chart ;  (4)  the  composite  track, 
where  wanted,  can  be  obtained  by  a  simple  construction ;  and  (5)  the 
distance  from  point  to  point,  if  wanted,  can  be  easily  determined. 

Id  the  stereographic  projection  of  the  sphere  the  point  of  projection 
O,  Fig.  1,  is  on  the  surface  of  the  sphere,  at  the  extremity  of  a  diameter 
PCO,  through  P,  the  center  of  projection.  Thus,  if  f?P/  represent  the 
tangent  plane  through  P  the  points  A  and  B  on  the  sphere  would  be 
projected  on  dPf  at  a  and  b,  where  OA  and  OB  produced  meet  dPf.  If 
D  and  E  are  90  degrees  from  P  (as  in  Fig.  1)  their  projections  fall  on 


Fig.  1. 


dFf  at  d  and  e.     A  j)oint,  as  F,  still  nearer  to  O,  will  be  projected  on  the 
tangent  plane,  as  at  /'. 

If  P  be  either  i)ole  the  projection  of  the  sphere  on  the  plane  diy  is 
a  very  simi)le  matter ;  for  all  the  meridians  are  i)rojected  into  straight 
lines  through  P,  and  all  the  latitude-parallels  into  circles  around  P  as 
center.  The  radii  of  these  circles  can  be  obtained  by  construction,  as 
shown  in  Fig.  i.  But  in  practice  it  is  far  better  to  use  their  known 
lengths,  as  indicated  in  trigonometrical  tables.  Thus,  if  PB  is  an  arc 
of  60°  we  know  that  the  angle  POB  contains  SO'^,  so  that  P6  is  equal 
to  PO  tan  30*^.  Thus,  for  the  parallels  corresponding  to  latitudes  85°, 
80°,  75°,  and  so  on,  we  take  from  the  trigonometrical  tables  the  natural 
tangents  of  2^ci,  5°,  7.^°,  and  so  on  ;  and  these  numbers,  with  any  con- 
venient unit  of  length,  give  us  the  radii  of  the  circles  we  are  to  describe 
round  P.  For  instance,  if  we  wish  the  equator  to  have  a  radius,  Pe, 
(equal  to  PO)  5  inches  in  length,  we  draw  a  line  5  inches  long,  divide  it 
into  ten  equal  i)arts,  and  one  of  these  again  into  ten  parts  (or  preferably 
we  make  a  plotting  scale  for  the  smaller  divisions);  then,  regarding  one 


47 

of  the  tenths  of  the  line,  /.  e.,  one-half  an  inch,  as  our  unit,  we  take  for 
our  successive  r.Klii  lines  having  the  following  lengths: 

For  latitude  85^,  0.437,  which  is  the  tangent  oi"  2^° 
For  latitude  80°,  0.875,  which  is  the  tangent  of  ~)^ 
For  latitude  75°,  1.317,  which  is  the  tangent  of  7i° 

and  so  on.  Then  a  series  of  radial  lines  drawn  to  divisions  5"^  apart 
round  any  one  of  these  circles  give  the  im-iidians,  and  complete  our 
l)rojectii)n.  The  chart  should  have  outlines  of  continents,  islands,  etc., 
marked  in  for  convenience,  though  in  reality  this  is  not  essential,  be- 
cause the  longitudes  and  latitudes  of  i)orts  and  i)laces  are  alone  really 
needed  for  determining  the  great  circle  track  ;  and  the  track  obtained 
by  the  simple  constructions,  which  will  be  indicated,  could  always  be 
plotted  in  on  the  Mercator  chart,  to  the  use  of  which  seamen  are  more 
accustomeil  than  to  that  of  any  other  kind  of  chart. 

The  properties  of  the  stereographic  projection,  which  enables  us  at 
once  to  project  a  great  circle  course  and  to  determine  bearings,  dis- 
tances, etc.,  on  a  stereographic  chart,  are  the  following  : 

(rt)  Every  circle  on  the  sphere,  great  or  small,  is  projected  into  a 
circle. 

(h)  All  angles,  bearings,  etc.,  on  the  sphere  are  correctly  presented 
in  the  projection  (a  property  found  a'so  in  Mercator's  i)rojection). 

With  these  properties  are  combined  the  following  properties  of  great 
circles  on  the  sphere  : 

(1)  Since  every  diameter  of  a  great  circle  is  a  diameter  of  the  sphere 
each  i^oiut  on  a  great  circle  is  antipodal  to  another  point  on  the  same 
circle,  or,  in  other  words,  if  a  great  circle  passes  through  any  point  it 
passes  also  through  the  antipode  of  that  point. 

(2)  If  a  great  circle  touches  a  small  circle  on  the  sphere,  it  touches 
also  the  small  circle  antipodal  to  the  former;  for  instance,  if  a  great 
circle  touches  latitude  parallel  30°  north  it  touches  also  latitude  par- 
allel 30°  south.  This  needs  no  demonstration,  being  really  a  corollary 
of  (1) ;  for  the  point  in  which  the  great  circle  touches  one  small  circle 
has  for  its  antipode  a  corresponding  point  on  the  antipodal  small  circle, 
and  also,  by  (l),on  the  great  circle;  and  there  can  be  no  other  jjointin 
which  the  great  circle  meets  the  antipodal  small  circle,  for  if  there 
were  then,  b}^  (1),  there  wo  uld  be  corresponding  points  of  contact  or  of 
intersection  with  the  original  small  circle,  which,  by  our  hypothesis,  is 
not  the  case. 

To  use  the  charts,  however,  the  seaman  need  not  concern  himself 
either  with  the  method  of  constructing  them  or  with  the  principles  on 
which  their  use  in  great  circle  sailing  depends.  All  he  need  care  for  is 
rightly  to  apply  the  constructions  which  result  from  these  principles. 

It  is  proposed  to  indicate  only  what  are  the  processes  necessary  for 
the  five  following  problems  : 

I.  To  Jind  the  great  circle  track  between  any  two  j^ioincs,  as  A  and  13, 
Fig.  2. 


48 

II.  To  find  the  vertex  V,  or  highest  latitude  reached  on  that  track. 

III.  To  find  the  hearing  at  any  pointy  as  Q,  on  the  track. 

IV.  To  find  the  composite  track,  from  port  to  port,  touching  any  given 
limiting  latitude. 

Y.  To  find  the  distance  AVB  to  be  traversed. 

The  constructions  for  these  five  problems  are  all  included  in  tlie  fol- 
lowing simple  statements  (P,  Fig.  2,  is  the  pole,  e'EE'  the  equator),  the 


D 


Fig.  2. 

italicised  parts  indicating  the  actual  constructions,  the  rest  giving  the 
reasons  and  demonstrations : 

I.  Find  on  the  chart  a  and  h  the  antipodes  of  A  and  B  (which,  of  course, 
is  easy,  as  we  have  only  to  take  ofl'  180°  on  the  meridians  APa,  BP6, . 
not  shown  in  the  figure,  to  avoid  crowding) )  then  a  circle  through  any 
three  of  the  points,  A,  a,  B,  b,  is  a  great  circle  of  the  sphere  by  (1),  and  must 
pass  through  the  fourth  point.  Describe  such  a  circle  AB  ah;  AVB  is  the 
great  circle  course  required.  [It  will  be  best  to  take  the  three  points  A, 
B,  and  b,  A  being  the  point  of  departure.  Pencil  the  bisecting  perpen- 
diculars to  6B,  AB  (not  shown  in  Fig.  2),  intersecting  in  C,  around 
which  point  as  a  center  describe  in  i)encil  the  circle  AB&.  JSTote  whether 
it  passes  through  a,  for  this  serves  as  a  test  of  the  accuracy  of  the  re- 
sult. You  may  also  note  whether  the  points  e'  and  E',  in  which  it  cuts 
the  equator,  are,  as  they  should  be,  on  the  extremities  of  a  diameter 
through  P  ;  for  two  great  circles  on  a  sphere  necessarily  intersect  on  a 
diameter  of  the  sphere,  and  therefore,  as  P  is  the  projection  of  the  pole 
of  the  equator,  two  such  points  of  intersection  must  lie  on  a  straight 
line  through  P.] 

\l.  A  straight  line,  vVQY,  through  P  and  C,  cuts  the  great  circle  track 
in  V,  the  vertex  required,  V  being  the  highest  latitude  on  one  side  of 
the  equator,  v  that  on  the  other.  [V  does  not  necessarily  fall  on  the 
actual  great  circle  course  between  two  points.  For  instance,  QB  is  the 
great  circle  course  from  Q  to  B,  but  V  lies  outside  Qli.j 

III.  Draw  QT,  tangent  to  the  track,  at  Q.  Then  the  angle  PQT  gives 
the  bearing  of  the  track  at  Q  from  the  due  northerly  direction  QP.     By 


49 

observing  that  QT  is  at  right  angles  to  CQ  we  can  get  the  bearing 
without  actually  drawing  QT.  Thus,  in  the  case  illustrated  in  the 
figure,  the  direction  (^T  is  north  of  due  east  by  an  angle  equal  toCQP, 
easily  measured  with  a  protractor. 

IV.  Suppose  AVB,  Fig.  3,  the  great  circle  course,  to  have  its  vertex 
V  in  inconveniently  or  dangerously  high  latitudes.  Let  L  on  W  be  the 
highest  latitude  which  the  ship  must  reach.  Take  I,  antipodal  to  L ;  then,  if 
wc  bisect  Ih  in  G,  and  describe  round  P  as  center  the  circle  cGc',  it  is 
obvious  that  any  circle  having  its  center  on  cGc',  and  radius  equal  to 
GL  or  G/,  will  touch  both  the  latitude  parallels  HLKand  klh;  and,  by  2, 


Fig.  :$. 


will  be  a  great  circle.  Therefore,  around  A  and  B,  as  centers,  icith  radius 
GL,  describe  circles  cutting  cGc'  in  c  and  c';  and  icith  c  and  c'  as  centers  and 
the  same  7-adius  describe  circular  arcs,  AH,  BK,  touching  the  latitude  par- 
allel ULK  in  Hand  K.  Then  A II  KB  is  the  composite  track  required. 
The  distances  along  AH  and  BK  can  be  determined  by  the  method 
shown  in  the  succeeding  section,  and  the  distance  along  the  latitude 
parallel  HLK  is,  of  course,  easily  determined,  being  an  arc  of  a  known 
number  of  degrees  in  a  known  latitude. 

V.  [We  have  to  determine  how  many  degrees  there  are  in  the  arc 
AVB,  not  as  it  appears  in  the  chart,  but  as  it  really  is  on  the  sphere.] 
Take  p,  9()o  of  latitude,  from  V  and  v,  Fig.  2,  along  vW.  (This,  of 
course,  is  done  at  once  on  the  chart,  which  shows  the  degrees  of  lat- 
itude from  the  pole  P.)  Then  p  is  the  pole  of  AVB.  Find  D,  the  cen- 
ter of  the  great  circle  Ap ;  and  F,  the  center  of  the  great  circle  Bp. 
(This  we  do  by  I. ;  but  most  of  the  work  is  already  done.  We  have  the 
bisecting  i)erpendicularto  B/> ;  the  bisecting  perpendicular  to  Aa  passes 
through  C;  then  the  bisecting  perpendiculars  to  A;;,  pB  give  us,  by 
theii-  intersections  with  those  to  Aa  and  B6,  D  and  F  at  once.)  What 
we  want  is  to  determine  the  angle  Aj>B,  between  the  arcs  A/>,  /^B  ;  and 
it  is  obvious  that  this  is  the  supplement  of  the  angle  V>p^,  which  is  easily 
19862 4 


50 

measured  with  a  protractor.  The  number  of  degrees,  multiplied  hy  60, 
gives  the  number  of  geographical  miles  or  hnots  in  the  distance  AVB. 

An  example  of  these  methods  is  given  in  the  stereographic  chart, 
which  has  been  reduced  from  Proctor's  great  circle  sailing  chart  for  the 
Soutlieru  Hemisphere.  In  this  chart  the  same  letters  are  used  as  in 
Figs.  2  and  3.  * 

SPHERICAL    TRAVERSE    TABLES. 

Under  the  methods  of  Bergen  and  Towson  the  general  computation 
of  the  parts  of  spherical  triangles  and  the  arrangement  of  the  results 
in  the  form  of  spherical  tables  were  treated  of.  In  liaper's  excellent 
work  on  navigation  spherical  traverse  tables  are  given.  Tliey  are 
adapted  to  great  circle  sailing  in  the  same  manner  as  the  ordinary 
traverse  tables  are  adapted  to  sailing  on  a  rhumb  line.  Practical  rules 
covering  all  cases  are  there  given  for  the  guidance  of  navigators. 

GREAT   CIRCLE    COURSES   FROM   THE    SOLAR   AZIMUTH   TABLES. 

At  page  8  of  a  treatise  on  Azimuth  ^  by  Lieut,  Commander  Joseph 
Edgar  Craig,  U.  S.  Kavy,  the  following  equation  is  stated  for  the  solu- 
tion of  the  time-azimuth  problem : 

,  rr     COS  Li  tan  d— sin  Li  cos  t  ,-. . 

cot  n  = -. — ; (1) 

sin  < 

from  which  are  derived 

tan  ^  =  co8  /.  cot  d  (2) 

and  cot  Z  =  cot  t.  cos  (^-f  Li)  .  cosec  (/>  (3) 

in  which  /,  d,  and  Z  represent  respectively  the  hour-angle,  declination, 

and  azimuth  of  the  observed  celestial  body,  and  Li  the  geographical 

latitude  of  the  observer. 

At  page  54  of  the  present  work  on  the  Development  of  Great  Circle 
Sailing,  the  equations  stated  for  finding  the  great  circle  course  are: 

tan  (p  =  cos  A  cot  Lj  (4) 

and  cot  C  =  cot  A  cos  (L,  +  ^)  cosec  (/>  (5) 

in  which  Li  represents  the  latitude  of  the  ship,  L^  the  latitude  of  the 
l)lace  of  destination,  A  the  difference  of  longitude  between  the  meridian 
of  the  ship  and  the  meridian  of  the  place  of  destination,  and  C  the 
course.  If,  in  equations  (2)  and  (3),  A  be  substituted  for  t  and  L^  for  d, 
their  second  members  will  be  identical  with  those  of  ecpiations  (4)  and 
(5),  and  therefore,  when  the  difference  of  longitude  and  the  latitude  of 
the  place  of  destination  in  the  great-circle  problem  are  equal  respec- 
tively to  the  hour-angle  and  declination  in  the  time-azimuth  ])roblem; 
the  course  C  resulting  from  equation  (5)  will  be  identical  with  the 
azimuth  Z  resulting  from  equation  (3),    Now  the  values  of  Z  resulting 


'  Azimuth.  A  Treatise  on  this  Subject,  with  .i  study  of  the  Astronomical  Triangle, 
and  of  the  Effect  of  Errors  in  the  Dat.a.  Ilhistrated  by  Loci  of  Maxininra  and  Mini- 
mum Errors.  By  .Joseph  Edgar  Craig,  Lieutenant-{;oinniandcr,  U.  S.  Navy.  New 
York :  John  Wiley  &  Sous,  1887. 


Chart  for  great  circle  sailing,  sho 
the  grea 


Face  page  50. 


51 


from  equation  (3)  have  been  computed  for  declinations  between  23°  N. 
and  23°  S.,  which  represent  the  range  of  declinations  of  the  sun,  and 
arranged,  for  stated  values  of  the  hour-angle,  declination,  and  latitude, 
in  the  Solar  Azimuth  Tables,  whicli  have  been  well  known  among 
marijiers  for  many  years.  It  is  therefore  evident  that,  when  the  i)ort 
of  destination  is  situated  within  the  Tropics,  the  Solar  Azimuth  Tables 
may  be  used  for  ascertaining  great  circle  courses  by  simply  regarding 
the  latitude  of  the  port  hound  to  as  declination,  and  the  difierence  of 
longitude,  converted  into  time,  as  the  hour-angle.  The  latitude  of  the 
shi})  remains  the  latitude  of  the  observer  as  in  taking  out  values  of 
the  azimuth  from  the  tables. 

The  identity  of  the  time-azimuth  problem  and  the  great-circle  course 
problem  may  also  be  graphically  illustrated. 


j-i^j. 


Fiff.2. 


Let  Fig.  1  represent  the  astronomical  triangle,  I*  ~  X,  projected  on 
the  plane  of  the  celestial  meridian  of  the  observer,  P^  P'  z' ^  and  let  P 
represent  the  elevated  pole,  ~  the  zenith  of  the  observer,  whose  latitude 
is  Li,  and  X  the  position  of  the  observed  celestial  body,  whose  declina- 
tion is  fZ,  hour-angle  #,  and  azimuth  Z ;  and  let  Fig.  2  represent  a  pro 
jection,  on  the  plane  of  the  terrestrial  meridian,  P^  P'  ~',  passing  through 
the  point  of  departure,  of  the  sidierical  triangle  P  ^  X,  whose  ver- 
tices are  the  place  of  departure  ~,  the  place  of  destination  X,  and  the 
elevated  i)ole  P,  and  whose  sides  are  arcs  of  the  meridian  of  the 
place  of  departure,  of  the  meridian  of  the  place  of  destination,  and  of 
the  great  circle  i^assing  through  the  places  of  departure  and  destina- 
tion. Then  it  will  be  apparent  that  if  the  values  of  L,  are  identical  in 
the  two  figures,  and  if  A,  the  difference  of  longitude  is  equal  to  #,  the 
hour  angle,  and  L.,,  the  latitude  of  the  place  of  destination,  is  equal  to 
d^  the  declination,  the  great-circle  course  C  must  be  identical  with  the 
azimuth,  Z. 

Captain  Craig,  the  author  above  mentioned,  and  others  of  the  fore- 
most navigators  of  the  Tnited  States  Navy,  had  noticed  the  ready 
adaptability  of  the  Solar  Azimuth  Tables  to  the  finding  of  great-circle 
courses,  and  had  for  years  made  use  of  the  knowledge  in  practical 


52 

navigatiou ;  but  no  formal  disclosure  of  the  method  appears  to  have 
been  made  uutil  refereuce  was  made  to  it  in  the  ninth  edition  of 
Lecky's  work  on  navigation,  entitled  Wrinkles  in  Practical  Naviga- 
tion. It  was  the  impression  of  this  author,  as  the  Azimuth  Tables 
extend  only  to  23  degrees  of  declination,  that  this  method  would  only 
be  aiiplicable  where  the  latitude  of  the  place  of  destination  is  within 
23  degrees  of  the  equator,  or  within  the  Tropics.  It  will  be  valuable, 
therefore,  to  point  out  to  navigators  that  the  Solar  Azimuth  Tables 
are  universally  applicable  for  finding  great-circle  courses  with  very 
great  facility,  because  all  great  circles  pass  into  the  Tropics;  and,  if 
the  problem  of  finding  the  courses  is  with  reference  to  a  great-circle 
track  between  a  point  of  departure  and  a  point  of  destination,  both 
lying  outside  of  the  Tropics,  it  is  only  necessary  to  find  a  iioint  lying 
on  the  prolongation  of  the  great-circle  arc  beyond  the  point  of  actual 
destination  and  within  the  Tropics,  and  treat  this  point  as  the  place  of 
destination  in  finding  the  courses  from  the  Azimuth  Tables.* 

To  illustrate,  take  the  problem  of  finding  the  initial  course  on  a  voy- 
age from  Bergen,  in  latitude  60"^  N.  and  longitude  5°  E.,  and  the  Strait 
of  Belle  Isle,  in  latitude  52°  12'  N.  and  longitude  55°  W.  On  a  copy 
of  a  gnomonic  chart,  such  as  Godfray's,  which  accompanies  this  work, 
draw  a  straight  line  between  the  geographical  positions  above  stated 
and  extend  it  beyond  the  latter  into  the  Tropics.  It  will  be  found  to 
intersect  the  twentieth  degree  parallel  of  latitude  in  longitude  90°  W., 
or  9.5°  from  the  meridian  of  the  point  of  departure.  Entering  the  Azi- 
muth Tables  at  latitude  60°,  under  declination  20°  and  opposite  hour- 
angle  95°  or  6'^  20'",  we  find  the  required  course  to  be  N.  75°  31'  W. 

*See  "The  '  Ex-Meridian '  treated  as  a  problem  in  Dynamics,"  by  H.  B.  Goodwin, 
M.  A.,  formerly  examiner  in  nautical  astronomy  at  the  Royal  Naval  College,  Green- 
wich, England.     London :  George  Philip  &.  Won,  32  Fleet  street,  E.G.,  1894. 


i 


SECTION  IV. 


METHODS  REQUIRING  COMPUTATIJN. 

This  section  is  devoted  to  the  exposition  of  those  methods  for  find- 
ing great  circle  courses  and  distances,  and  the  latitudes  and  longitudes 
of  points  on  great  circular  arcs  which  require  computation. 

THE   COMPUTATION   OF   GREAT   CIRCLE   DISTANCES. 


A'^ 


^ r^ 


Let  AB  represent  the  arc  of  a  great  circle  passing  through  the  two 
points  A  and  B,  whose  difference  of  longitude  is  A  and  whose  latitudes 
are  (p\  and  tp-i ,  respectively.  And  let  d  be  the  distance  between  A  and 
B,  measured  along  the  arc  AB.  From  the  fundamental  equations  ot 
spherical  trigonometry  we  have,  in  the  triangle  ABP, 

cos  d=cos  (90O— <pi)  cos  (900—^2)+ sin  (90O— 9>i)  sin  (90O— ^2)  cos  A 
cos  (Z=sin  q)x  sin  (7>2+cos  q)\  cos  (pi  cos  A 
=sin  cpi  sin  cpi  (1+cot  cp^  cot  cp-i  cos  A) 

THE  COMPUTATION  OF  GREAT  CIRCLE  COURSE  AND  DISTANCE,  AND 
THE  LATITUDES  AND  LONGITUDES  OF  POINTS  ON  GREAT  CIRCU- 
LAR ARCS. 


Fig.  1. 

Let  WE  represent   the  equator,  A  and  B   the  gix^en  points,  PA 
and  PB  the  meridians  passing  through  A  and  B,  respectively,  and 

53 


54 

AB  the  required  arc.  Let  BF  represent  the  great  circle  drawn  from 
one  of  the  given  points  perpendicular  to  the  meridian  passing  through 
the  other.  Let  A  =A.PB  denote  the  difference  of  longitude  between  A 
and  B. 

Let  Li  represent  the  latitude  of  A, 
L2  the  latitude  of  B, 

qj  the  arc  PP. 

Then  AF  will  equal  90° — (L,  +  97),  and  we  have — 

tan  cp  =  cos  A  cot  L2  '  (1) 

cot  C  =  cot  A  cos  (Li4-^)  cosec  qt  (2) 

cot   f?  =  cos  O  tan  (Li  +  <^)  (3) 

in  which  C  represents  the  course  from  A;  but,  should  the  course  from 
B,  be  desired  iutercbange  Li  and  L^  in  each  of  the  above  formultB.  In 
drawing  the  diagram,  let  fall  the  perpendicular  upon  the  meridian 
passing  through  that  point  from  which  the  course  is  desired. 

The  signs  of  the  functions  must  be  carefully  noted.  That  branch  of 
the  great  circle  which  corresponds  to  A  <  180°  is  sought. 

cp  may  be  taken  in  the  first  or  second  quadrant,  according  to  the  sign 
of  its  tangent;  but  it  will  be  found  convenient  to  restrict  it  to  a  value 
less  than  90°,  marking  it  positive  or  negative,  according  to  the  sign  of 
its  tangent. 

The  latitude  of  one  of  the  points  being  regarded  as  positive  that  of 
the  other  point,  when  of  the  opposite  name,  must  be  marked  negative. 
Thus,  in  Pig.  2,  PB  is  numerically  90°+ L2. 


To  find  the  position  of  the  vertex,  iu  the  triangle  PAC*>  (Fig.  1),  C" 
being  the  vertex, 

cos  Lv=siu  C  cos  L,,  (4) 

cot  Ay  =tan  C  sin  Li,  (5) 

If  the  angles  PAB  and  PBA  are  both  less  than  90°,  the  vertex  will 
be  between  the  points  A  and  B ;  but  if  the  course  from  one  given  point 


55 


is  less  than  QO'^  and  that  from  the  other  greater  than  90°,  the  vertex 
will  be  upon  the  arc  AB  extended.  The  latitude  of  the  vertex  sought 
will  be  of  the  same  name  as  that  of  the  given  point,  which  is  nearer  to  a 
Xiole  than  the  other  point. 

To  find  other  points  of  the  curve,  as  C  and  0",  assume  differences  of 
longitude  from  ths  vertex,  usually  at  intervals  of  5*^  or  10°;  Ihen 


tan  L'=tan  L^  cos  A/ 
tan  L"  =  tan  L„cos 


>s  A/ ) 
s  X"  \ 


(G) 


Each  of  these  forrauhe  will  determine  two  points  of  the  curve,  symmet- 
rically situated  on  opposite  sides  of  the  vertex;  bat  only  one  of  these 
will  be  Uiied  when  the  vertex  falls  outside  of  the  required  arc. 

Likewise  successive  values  of  the  latitudes  may  be  assumed,  and  the 
corresponding  differences  of  longitude  found  by  the  formuhe : 


cos  A'=  cot  L, 
cos  A"=  cot  L, 


tan  L' 
tan  L 


;.! 


{-: 


ASMUS'S  METHOD    FOR   THE   CONSTRUCTION   OF   A    GREAT   CIRCLE    ON 
THE   MERCATOR   PROJECTION. 


Fig.  1. 

Let  ABMN  (Fig.  1)  represent  the  equator,  C  the  center,  k.C=(i  the 
radius,  and  D  the  pole  of  the  earth.  Let  AD  represent  the  prime 
meridian,  as  for  instance  the  meridian  of  Greenwicli ;  MI),  any  other 
meridian;  the  angle  ACM=A,  the  longitude;  the  angle  MCP=^,  the 
latitude  of  the  point  P. 

Let  BP  represent  a  great  circle  cutting  the  equator  at  an  angle 
MBP  =  ;)/,  and  let  the  point  of  intersection  B  be  determined  by  its  longi- 
tude, /^,  the  angle  ACB. 

Suppose,  now,  that  BP  takes  the  infinitesimal  increment  PQ,  then  A 
will  be  increased  by  the  angle  MCN  =  fZA,  and  q)  by  the  angle  \lCQ,—d(f). 
PK  denotes  a  part  of  the  parallel  passing  through  P.  The  radius  of 
the  parallel  PR  is  a  .  cos  cp. 


56 

Since  PR  is  described  with  the  radius  a  .  cos  9>,  and  belongs  to  the 
central  angle  ^A,  the  arc  PR=a  .  cos  q)  .  dX,  and  since  QK  is  described 
with  the  radius  a,  and  corresponds  to  the  central  angle  dcp, 

QR=a  .  d(p. 

Since  infinitesimal  arcs  may  be  considered  straight  lines  it  follows 
that  the  triangle  PQR  may  be  considered  a  plane  one. 

In  the  Mercator  projection  the  lines  PR  and  MN  (Fig.  1)  are  equal  to 
each  other.  Since  this  is  really  not  the  case,  and  since  it  is  desirable 
that  the  spherical  figure  and  its  representation  be  similar,  if  not  as  a 
whole  yet  in  their  smallest  parts,  and  since  it  is  necessary  that  the 
angle  QPR  be  the  same  on  the  sphere  and  the  chart,  w^e  imagine,  instead 
of  the  triangle  PQR  a  similar  one,  P'Q'R'  (Fig.  2),  whose  sides  are  sec  q) 
times  as  large  as  those  of  PQR.    Therefore,  in  the  triangle  P'Q'R', 


Fig.  2. 


P'R'=PR  sec  q)=a  .  cos  <p  .  dX  .  sec  <^=a  .  dX 
which  makes  P'R'=MN, 


Q'R'=QR  sec  q)=a  .  dcp .  sec  q)-=a  . 


dcp 
cos^ 

In  Fig.  2,  AB,  MN,  and  AM  are  the  same  as  in  Fig.  1. 

AB  =  a./? 

AM  =  a  .A 

MN  =  a.dA 

Moreover,  the  triangle  P'Q'R'  is  similar  to  the  triangle  PQR.  If  the 
equator  be  taken  as  the  axis  of  X,  and  the  prime  meridian  as  the  axis 
of  Y,  we  have  : 

x  =  AM  =  a.X  (1) 

and  if  the  unknown  ordinate  M  P'  be  denoted  by  y, 


dy  =  Q'R'  =  a  .  dcp  .  sec  cp 


_  a.  dcp 
~  COScp 


57 

and  by  integration, 

y  =  a  loiT  iiat.  tan  (^  +  'f')  ^2) 

A  tliird  equalioii  is  needed  which  shall  express  that  the  great  circle 
BP  (Fig".  1)  cuts  the  equator  at  an  angle  ;/. 

From  the  spherical  triangle  BMP,  two  of  whose  sides  are  (A  —  (i)  and 
(p,  we  have 

tan  ;'  sin  (A  —  p)  =  tan  (p  (3) 

Knowing  a,  /J,  and  ;/,  and  giving  x  any  value,  we  obtain  A  from  equa- 
tion (1),  then  (/v  from  (3),  and  linally  y  from  (2). 

In  order  to  express  the  curve  BP'  (Fig.  2)  only  in  rectangular  co-ordi- 
nates, an  equation  which  does  not  contain  the  variables  A  and  (p  must 
be  deduced  from  (1),  (2),  and  (3), 

From  (1), 
from  (2), 


a 


or  applying  the  formula 

,        o£.        2  tan  6  .,  „  ^     /'tt  ,  q)\ 

*^"  ^' =i::::ton^r,    to  the  case  of  e={j+^) 

y 

tan  (^  4-  ^  )  =  -^=  -  cot  cp 


1—e  « 

iy  y        —y 

ga  —1     e^-c" 


tun  cp=^^ = —  (5) 

2e« 

Finally,  substituting  the  values  of  A  and  tan  (p  from  (4)  and  (5)  in 
equation  (3),  we  obtain  as  the  equation  to  the  curve  B  P' 

X         ,,     \       6"— 6" 


tan  ;/  sin  (  ^  -/i  J  =  — ^—  (6) 

Up  to  this  point  /?  and  y  have  been  assumed  as  known.  If  this  is  not 
the  case,  and  we  have  instead  the  conditions  that  the  curve  must  pass 
through  two  points  whose  latitudes,  cpi  and  (p-i,  and  longitudes,  Ai  and 
Az,  are  given,  then  /i  and  ;/  must  be  computed  by  the  two  following 
equations,  which  are  obtained  from  (3) : 

tan  ;/  sin  (Ai  — /^)=tau  cpi 

tan  y  sin  (Aj— /i^)=tan  cpt  (7) 


58 


The  quotient  of  the  equations  (7)  contains  but  one  unknown,  which  is 
readily  found  by  following  out  the  operations  which  are  indicated  below  : 


tan  (^-^^'-6  V«|M^!+^^ .  tan  ^^lAi^tan  6 
V     2        ^    y     sin  {(p-2-(pi)  2 


in  which  d  is  an  auxiliary  angle. 
Having  found  6, 

Knowing  /?,  we  have  from  (7), 


3^2+^1  _^ 


tan  r—     ^^"^'       -     ^^°^^ 
I..IU  y-  ^.^  (Aj_/i)     sin  [Xo-fi] 


(8) 


(9) 


(10) 


In  order  to  construct  the  curve  accurately,  /i  and  ;/  are  first  deter- 
mined from  (9)  and  (10),  then  (1),  (2),  and  (3)  are  applied  to  a  series  of 
longitudes  between  Ai  and  A2,  the  extreme  longitudes  of  the  track. 

In  the  case  in  which  the  terminal  points  Aj  q)i  and  A2  cp-i  are  not  far 
apart  the  following  approximate  construction  may  be  applied.  For  the 
point  of  departure  we  have  at  once, 

Xi=a\i  yi=«lognattan  (  ^4-^M  (11) 

and  for  the  point  of  destination, 


X2=aX2  2/2=*lognat  tan  ^  J-f-^M 


(12) 


Since  the  straight  line  between  these  two  points  would  depart  too 
much  from  the  true  curve  it  is  better  to  connect  them  by  an  arc  of  a 
circle,  to  determine  which  a  third  point  is  necessary.  This  may  be  most 
easily  obtained  by  determining  the  direction  of  the  tangents  at  the  ter- 
minal points  of  the  curve  Pi  and  P2  (Fig.  3). 


Fig.  3. 


59 

Let  T  denote  the  tangent  angle=angle  RPQ,  then 

~  P'K'  ~  Pli  ~  cos  cp  '  dX 
By  differentiating-  e<iuation  (3),  we  have 

tan  y  cos  (A  — /i)  dX  =  — ^— 

J^=  tan  j/  cos  (A  — /:^)  cos^  y> 
Whence 

tan  T=tan  y  cos  (A— /i)  cos  a^,  or  since  by  (3)  tan   y=    .    '  '1  ^ — 
^  ^  •'   V  /  -^      sm  (A— ytf) 

tan  r  =  cot  (A—/:/)  sin  ^  (13) 

which  does  not  contain  y. 

The  positions  of  the  tangents  at  Pi  and  P2  are  now  determined  by  the 
formulae 

tan  Ti  =  cot  (Aj— /?)  sin  (pi  and 

tan  T-z  =  cot  {A-,—/3)  sin  cpz 

With  these  two  tangents  two  circular  arcs  may  be  constructed  passing 
through  the  points  Pi  and  P2.  The  centeus  Oi  and  O3  of  these  circles- 
are  the  intersections  of  the  normals  at  Pi  and  P2  with  the  perpendicular 
to  the  middle  point  of  the  chord  joining  Pi  and  P2.  The  real  curve  is 
contained  in  the  meniscus  between  the  two  arcs,  and  may  be  drawn  suf- 
ficiently close  with  an  irregular  curve. 

This  method  is  too  tedious  for  purposes  of  navigation,  and  if  used  at 
all  by  the  navigator  it  will  be  sufficient  to  determine  the  successive  posi- 
tions through  which  the  great  circle  passes,  to  plot  these  positions  upon 
the  chart,  and  connect  them  by  circular  arcs.  The  courses  and  dis- 
tances may  then  be  taken  directly  from  the  chart.  As  far  as  the  deter- 
mination of  the  positions  of  the  successive  points  on  a  great  circle  is  con- 
cerned this  method  is  shorter  than  those  ordinarily  emploj'ed.  It 
originated  with  J.  Asmus,  of  the  Hydrographic  office  of  the  German 
Empire,  and  was  published  in  the  Annalen  der  Hydrographie  und  ma- 
ritimen  Meteorologie,  1879,  Vol.  lY,  p.  151  et  seq. 

ZESCEVICH'S.  METHOD   FOR  FINDING  THE   POSITIONS   OF   THE   POINTS 
OF   THE   ARC   OF   A   GREAT   CIRCLE. 

This  method,  which  was  published  by  Piofessor  Gelcich  in  the  "  Mit- 
theilungen  aus  dem  Gebiete  des  Seewesens,"  Nos.  IX  and  X,  188G,  is 
due  to  the  late  German  hydrographer  Zescevich.  It  consists  in  com- 
puting the  latitude  at  the  middle  longitude  of  a  great  circular  arc,  and 


60 

has  the  advantages  of  simplicity,  saving  of  time  in  calculation,  and  ex 
eluding  the  i)0ssibility  of  error  in  plotting. 


Let  A  be  the  point  of  departure  and  B  that  of  destination.  Let  ACB 
be  the  arc  of  a  great  circle  passing  through  A  and  B.  Let  P  be  the 
pole,  ^0  the  latitude  of  A,  and  cp^  the  latitude  of  B.  AP  is  therefore 
equal  to  90°  — ^o,  and  BP  equal  to  90°  — ^i.  Bisect  the  difference  of 
longitude  APB=JA,  by  the  meridian  PC  and  let  the  angle  PCB  be 
represented  by  x. 

Let  PC=2/,  AC=&,  and  BG=a,  then,  from  the  triangle  PBC, 

sin  (p\  =  cos  y  cos  a  +  sin  y  sin  a  cos  x  (1) 

cos  a  —  cos  y  sin  cpi  +  sin  y  cos  cpi  cos  ^z/A  (2) 

Substituting  the  value  of  f.os  a  from  equation  (2)  in  equation  (1).  we 
obtain : 

sin  (pi  sin  y  =  cos  y  cos  q)i  cos  ^z/A-|-  sin  a  cos  x 

and  dividing  by  cos  cpi 

.  .  1  ^1   ,  sin  a .  cos  x 

tan  £0i  sin  y  =  cos  y  cos  A^a  -\ — 

^  -^  -^         -  cos  q)i 

Since 

sin  a       sin  ^z/A 

cos  cpi  ~     sin  X 

tan  q)i  sin  y  =  cos  y  cos  ^/JX  4-  sin  ^ JA  cot  x  (3) 

Similarly  we  obtain  from  the  triangle  APC 

tan  q)^|  sin  y  =  cos  y  cos  J JA  —  sin  ^JX  cot  a?  (4) 

Adding  (3)  and  (4), 

sin  y  (tan  c/j,)  +  tan  ^,)  =  2  cos  y  cos  Jz^A 


or 


sinysin(yo+y>i)^2  cos  y  cos  .i  JA 
cos  q)Q  cos  ^i 


61 


or 


tan  V  —  2co8^^  j;i  cos  (pa  cos  ^  ,gv 

"~  sin  (y^-I-  '•A'l) 

2/  is  the  conii)lenieiit  of  the  hititiule  of  tlie  i)oiiit  C,  which  lies  on  the 
great  circle  e(iui(listant  in  longitude  l)etw<;en  the  two  i)oints  marking 
the  extremities  of  the  arc  in  question.  Jf  the  latitude  of  this  i)oiut  be 
denoted  by  (/; ,  then  (00°—  </;  )  =  i/,  and  if  this  value  be  substituted  in 
(5)  and,  at  the  same  time,  the  reciprocal  of  (5)  be  taken. 

tany>  =  ,         sin(c^o+y..) ^6^ 

i      L'  cos  ^/jA  cos  qjo  cos  (pi 

In  this  equation  the  sign  of  the  cp^  will  depend  upon  the  sign  of  the 
numerator.  If  north  latitude  be  denoted  by  +  and  south  by  —  the 
resulting  latitude  cp  will  be  north  when  the  algebraic  sum  of  cp^  and 
(pi  is  positiv  e  and  south  when  it  is  negative. 

The  latitude  of  that  point  on  the  great  circle  whose  longitude  is  equi- 
distant from  A  and  C  is  now  found  by  the  same  formula,  aud  similarly 
the  latitude  of  the  point  midway  in  longitude  between  C  and  B. 

According  to  the  adopted  notation  these  formuhe  will  be 


and 


s       3  cos  ^ZJA  cos  cpo  COS  cp^ 

t^n  cp,  =  - f'in^^^l 

2  cos  ^zJA  cos  q)i  cos  (^j 


The  latitudes  of  these  points  of  the  great  circle,  which  are  midway  in 
longitude  between  the  points  now  known,  can  be  computed  in  like  man- 
ner until  the  positions  of  the  requisite  number  of  points  are  established. 

In  computing  these  sets  of  formulic  many  of  the  logarithms  are  re- 
peated, aud  the  work  is  thus  materially  abridged,  as  may  be  seen  from 
the  following  set: 

-Proceeding  with  the  computation,  it  would  next  be  necessary  to  de- 
termine ^j,  (pg,  (pi,  and  q)i^  for  which  the  equations  are 


,  sin  ((po-\-  (Pi) 

tan  fflj  =  —- vy»-rv-4/ 

2  cos  ^z/A  cos  ^0  cos  (p^ 

t^u(p,=- — ^i^in^^^) — 

^     2  cos  ^JA  cos  (pi  cos  (p^ 

tan  cp,= smi(pi,  +  (pi) 

^^      2  cos  iJA  cos  (ph  cos  (pi 

.  sin  ((pi+  (pi) 

tan  (pi  =  ri r-A-i  -^ 

'     2  cos  J JA  cos  q>i  cos  (pi 


62 


THE    COMPUTATION    OF    THE  LATITUDE    AT   THE    MIDDLE    LONGITUDE. 

A  shorter  and  more  direct  method  of  arriving  at  Zescevich's  results 
Is  as  follows:  Let  the  great  circle  ab  be  gnomonically  projected  upon  a 
plane  tangent  to  the  sphere  at  a  point  O  on  the  equator,  as  indicated 
in  the  accompanying  figure,  in  which  the  plane  of  projection  is  repre- 
sented as  having  been  rebatted  into  the  plane  of  the  equator. 


^qtudor 


Let  OP  —  r  —  1  represent  the  radius  of  the  sphere, 
q)  the  latitude  of  the  point  «,  projected  at  rtj, 
cp'  the  latitude  of  the  point  &,  projected  at  i»2, 
q)^  the  latitude  of  the  point  w,  projected  at  m-i, 
A  the  difference  of  longitude  between  a  and  h. 
Then  of  =  og  =  tan  i  A 
ftti  =  sec  ^  A .  tan  cp 
(jhi  —  sec  h  A.  tan  ^^ 

om-i  —  tan  f/>i  =''       '^ 

—  i  sec  .]  A  (tan  (/>  +  tan  cp^). 
If  the  extremities  of  the  great  circular  arc  in  question  have  latitudes 
of  different  name,  the  formula  for  the  latitude  at  the  middle  longitude, 
(p  being  the  greater  latitude,  is  tan  (p  i  =  ^  sec  ^  A  (tan  q^  —  tan  </>,). 


63 


LIST  OF  LITEKATJRF.    (POX  THE  SFIUKCT  OF  (IRE.IT  CIRCLE  SAJLLVO. 

Aiinalen  der  llydrograpbie  uiid  iiiaiitimen  Meteorologie,  Vol.  XII, 
18S(),  p.  r>;5G;  also  Vol.  lY,  1871),  \).  151  et  seq. 

Azimuth  Tables  for  parallels  of  latitude  between  (M'^  N.  and  Gl^  S., 
and  Ibr  declinations  between  23'^  N.  and  23^'  S.,  by  Lieuts.  Seaton 
Schroeder  and  W.  H.  H.  Southerland,  U.  S.  N.,  4to.,  1897.  Hydro- 
graphic  Office  publication  No.  71. 

Einfoche  Losung-  der  Probleme  der  Schitlahrt  ini  grossteu  Kreise, 
Dr.  F.  Paugger,  Triest,  1805. 

Freucli  Hydrographic  Chart,  No.  3680,  Lieutenant  Hilleret,  French 
Navy. 

Great  Circle  Sailing  Chart  on  the  Gnoraonic  Polar  Projection,  Hugh 
Godfray,  M.  A.,  London:  J.  D.  Potter,  1858. 

Great  Circle  Sailing  IMade  Easy,  John  Seaton,  London,  1850. 

Great  Circle  Sailing:  indicating  the  shortest  sea  routes  and  describ- 
ing maps  for  finding  them  in  a  few  seconds,  by  Richard  A.  Proctor, 
London :  Longmans,  Green  »S:  Co.,  1888. 

Hydrographic  Office  Charts,  Nos.  904, 995, 1127, 1128, 1129, 1280, 1281, 
1282,  1283,  and  1284. 

Mittheiluugen  aus  dem  Gebiete  des  Seewesens,  Nos.  IX  and  X,  1886. 

Navigation  and  Nautical  Astronomy,  John  Kiddle,  F.  R.  A.  S,,  Lon- 
don :  Sinipkin  and  Marshall,  1857. 

Navigation  in  Theory  and  Practice,  Henry  Evers,  London  and  Glas- 
gow, 1875. 

Navigation  and  Nautical  Astronomy,  Merrifield  and  PWers,  London, 
1868. 

Notes  on  Navigation  and  the  Determination  of  Meridian  Distances, 
Commander  P.  F.  Harrington,  U.  S.  Navy,  Washington,  1882. 

Nouveau  calculateur  nautique  pour  efiectuer  rapidemeut  tons  les 
problemes  de  la  navigation,  E.  Lartigue,  Paris,  1884. 

Nouvelles  tables  destinees  a  abreger  les  calculs  nautiques,  Perrin, 
Enseigne  de  vaisseau,  Paris. 

Practice  of  Navigation  and  Nautical  Astronomy,  Lieut.  Henry  Kaper, 
E.  N.,  London:  J.  D.  Potter,  1866. 

Proceedings  of  the  United  States  Naval  Institute,  Vol.  XI,  No.  2, 
1885;  also  Vol.  XXIII,  No.  1,  1807. 

Spherical  Tables  and  Diagram,  with  their  application  to  Great  Circle 
Sailing,  W.  C.  Bergen,  London:  Simpkin  and  Marshall,  1857. 

Spherical  Tables  and  Diagram,  A.  H.  Deichnum,  London:  1857. 

Tables  for  Facilitating  Great  Circle  Sailing,  John  T.  Towson,  London: 
1850. 

Tables  pratiques  pour  la  navigation,  Bretel,  lieutenant  de  vaisseau, 
Paris,  1880. 

Tableau  des  distances  de  port  a  port.  Published  by  the  French  Gov- 
ernment, 1882. 

The  Theory  and  Practice  of  Great  Circle  Sailing  under  one  general 
rule,  with  examples,  liev.  P.  Robertson,  London:  Bell  and  Dalby,  1855. 

C 


% 


(y 


1^070     DATE  DUE       ^2070 


,.-,  .,         1 

w 

19*T3 

Rsnj  HCi 

V  29    1973 

nECl2i 

)74 

OCT  of  ^'  ' 

6  1974 

^no*? 

-«     A    100*^ 

hkct '■':'■ 

1  0  'V 

OCT 

1  6  198b 

-I  ■.  ,.  . 

GAYLORD 

PRINTED  IN  U.S.A. 

II  mil  III  III  I  I 


3   1970  00487   5891 


yQ  cfjiiTHfH'';  Rt':"'l'';i''L  [iBRAR/  ^i.rlLITY 


AA    000  596  299    8 


